                                                      SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION θ(x)

                                                  SAMUEL BROADBENT, HABIBA KADIRI, ALLYSA LUMLEY, NATHAN NG, KIRSTEN WILK


                                                   Abstract. In this article, we provide explicit bounds for the prime counting functions θ(x) for
                                                                                                                                         ck x
                                                   all ranges of x. The bounds for the error term for θ(x) − x are of the shape εx and (log     , for




arXiv:2002.11068v2 [math.NT] 27 Jan 2021
                                                                                                                                            x)k
                                                   k = 1, . . . , 5. Tables of values for ε and ck are provided.




                                                                                           1. Introduction
                                           1.1. History. In 1852 Chebyshev [5] proved that if x is large enough, then
                                                                            x                     x
                                                                   0.9212       ≤ π(x) ≤ 1.1056         as x → ∞,
                                                                          log x                 log x
                                           where π(x) denotes the number of primes less than or equal to x. It was the first major step towards
                                           the prime number theorem. He introduced what are now referred to as the Chebyshev functions:
                                                                             X                         X
                                                                     θ(x) =     log p, and ψ(x) =            log p,
                                                                                     p≤x                     pk ≤x,k≥1

                                           and he proved that for all x ≥ 30
                                                                                                                     2
                                                                   Ax − 52 log x − 1 < ψ(x) < 56 Ax + 4 log
                                                                                                         5              5
                                                                                                            6 (log x) + 4 log x + 1,
                                           and
                                                           1                                                              1
                                                              5           2                                      5           2
                                              Ax − 12                        15                   6                             5
                                                    5 Ax − 8 log 6 (log x) − 4 log x − 3 < θ(x) < 5 Ax − Ax + 4 log 6 (log x) + 2 log x + 2
                                                        2                                                  2

                                                               1   1   1   1
                                           where A = log(2 2 3 3 5 5 30− 30 ) = 0.9212 . . . and 56 A = 1.1055 . . . 1.
                                           As a consequence, there exists x0 > 0 such that
                                                                               0.9212x ≤ ψ(x) ≤ 1.1056x for all x ≥ x0
                                           and that there exists x1 > 0 such that
                                                                               0.9212x ≤ θ(x) ≤ 1.1056x for all x ≥ x1 .
                                           Such bounds are now known as Chebyshev bounds. Over the years, many other elementary argu-
                                           ments have yielded improved bounds:
                                                                               Bounds for ψ(x)/x      Bounds for θ(x)/x
                                                              Author
                                                                            upper    lower    range  upper lower range
                                                        Erdös (1932) [16] 1.38629     -      x>0      -       -      -
                                                       Hanson (1972) [22]      -       -        -   1.09861    -    x>0
                                                       Grimson & Hanson
                                                                           1.09861     -      x > 0 1.0508     -    x>0
                                                            (1977) [20]
                                                           Deshouillers    1.07715     -      x>0      -       -      -
                                                            (1977) [10]        -    0.92129 x ≥ 59     -       -      -

                                             2000 Mathematics Subject Classification. 11N05, 11M06, 11M26.
                                             Key words and phrases. prime number theorem, ψ(x), θ(x), explicit formula, zeros of Riemann zeta function.
                                             1There is a typo in the definition of A in [5] on p. 376. We have given a corrected definition of A.
                                                                                                   1
The Prime Number Theorem, as proven independently by de la Vallée Poussin [51] and Hadamard
[21] in 1896, states that the number of primes up to x satisfies
                                                   x
                                        π(x) ∼         as x → ∞.
                                                 log x
It can also be easily reformulated as

                                 ψ(x) ∼ x and θ(x) ∼ x as x → ∞.

The essence of the proof as suggested by Riemann is to relate ψ(x) to the zeta function ζ(s). This
allows one to use the properties of the zeros of ζ, and in particular their location in the complex
plane. For instance, Hadamard and de la Vallée Poussin proved that ζ(s) does not vanish on the
vertical 1-line. By refining this result, de la Vallée Poussin proved in 1899 that the error √ term
in estimating π(x) − li(x) (and θ(x) − x and ψ(x) − x) is asymptotically of size x exp(−c log x).
Between 1941 and 1976, Rosser and Schoenfeld (together or separately) developed a program of
determining explicit results for the Chebyshev functions, as well as various finite sums and products
over primes, including Mertens sums, and the size of the nth prime. We provide a sample of the
numerous inequalities they established:

                               0.980x ≤ θ(x) ≤ 1.019x, for all x ≥ e20 [40, Equation (12)],
                            x                         x
                                < π(x) < 1.25506           , for all x ≥ 17  [42, Corollary 1],
                          log x                    log x
                                       1                  1
     |ψ(x) − x|, |θ(x) − x| < x(log x) 2 exp − ( logR x ) 2 , for all x ≥ 2 [42, Theorem 11],
                                                   x
            |ψ(x) − x|, |θ(x) − x| < 0.0242269          , for all x ≥ 108     [43, Theorem 7]
                                               log x

with R = 17.51 . . .. Here is a non-exhaustive list that the interested reader can consult: Axler [1],
Büthe [4], Costa-Pereira [7], Dusart [15], Faber and Kadiri [17], and Trudgian [50].

1.2. Main Theorem. Our goal is to give a comprehensive and complete description of how to
obtain an explicit bound for the error term for θ(x) of the form (logxx)k , no matter the size of x and
for values of k that are most widely used.

Theorem 1. Let k be an integer with 0 ≤ k ≤ 5. For any fixed X0 ≥ 1, there exists mk > 0 such
that, for all x ≥ X0
                                                 
                                            mk
(1.1)                            x 1−               ≤ θ(x).
                                         (log x)k

For any fixed X1 ≥ 1, there exists Mk > 0 such that, for all x ≥ X1
                                                          
                                                     Mk
(1.2)                               θ(x) ≤ x 1 +              .
                                                  (log x)k

  In the case k = 0 and X0 , X1 ≥ e20 , we have
                                         −1/2      −2/3       −4/5
          m0 = ε(log X0 ) + 1.03883(X0          + X0       + X0      )   and   M0 = ε(log X1 ).

See Table 14 for values of m0 and M0 , and Table 15 for values of mk and Mk , for k ∈ {1, 2, 3, 4, 5}.

                                                       2
   Here ε(b) is a positive constant associated to ψ(x), defined in the next theorem.
The proof of Theorem 1 is given in Section 3. In addition, the reader will find there the first formal
algorithm to automatically deduce new bounds for θ(x) every time new bounds are generated for
ψ(x). In particular, we describe how the values for mk and Mk depend on ε(b). We have also
produced extended versions of Tables 8 - 15 in [2] which will be made available on our personal
webpages and on the arXiv.
   Büthe’s [4, Theorem 2, (1.7)] implies that, for all x < 1019 , θ(x) < x , giving Mk = 0 for all k
and x in this range. In addition we did direct calculations for values of x up to 7 · 1011 (see Table
13). Thus Theorem 2 gives relevant values for mk , Mk for X0 , X1 > 7 · 1011 > e27 as listed in Table
15. For more extensive calculations of mk , Mk , we refer the reader to Tables in [2]. For instance,
we obtain from [2, Table 14 and Table 15] respectively that, for all x ≥ 1019 ,


                                               θ(x) − x
                            1.9338 · 10−8 <             < 1.9667 · 10−8 and
                                                   x
                                              θ(x) − x    3.79 · 10−5
                                                        <             .
                                                  x         (log x)2


The following result gives explicit bounds for ψ(x) and is based on the articles [3], [4] and [37].

Theorem 2 (Büthe, Platt-Trudgian). Let b > 0. Then there exists a positive constant ε(b) such
that

                                    ψ(x) − x
(1.3)                                        ≤ ε(b), for all x ≥ eb .
                                       x

   We use [3, Theorem 2], [4, Theorem 1] and [37, Theorem 1] to compute a more exhaustive list of
values for ε(b) which we need for our calculations of mk , Mk . In particular details of the calculations
of ε(b) are provided in Appendix A and tables of values are given in Table 8 of Appendix B. The table
lists the best values obtained using either of these techniques. Büthe’s uses a smoothing technique
as introduced in [17] with a weight arising from the Logan function, while Platt and Trudgian use
a truncated Perron’s formula combined with the zero density obtained in [25]. Büthe’s technique
leads to better bounds when x < e2300 while Platt and Trudgian’s works better for larger values of
x. For instance for all x ≥ e3000 , the method from [37] gives ε(b) = 4.60 · 10−14 , and for all x ≥ e46 ,
method from [4] gives ε(b) = 6.95 · 10−9 . Improved estimates for ψ(x) may be used to derive new
bounds for θ(x). More precisely, we have


                                |θ(x) − x| ≤ |ψ(x) − x| + ψ(x) − θ(x),

                                                     √
where ψ(x) − θ(x) introduces an error term of size x. We study this term in Section 2, and
Theorem 5 provides a refinement to [43] and [15]. We give just below a non-exhaustive historical
recollection of bounds of the type (1.1) and (1.2). The values in Tables 1-5 make use of explicit
formula techniques.



                                                    3
                                           Table 1. Case k = 0

          Author                                       m0         X0         M0   X1
                                                         -          -    0.0376     1
                                                    0.0393      e13.8    0.0376 e13.8
          Rosser (1941) [40]
                                                    0.0328        e15    0.0321   e15
                                                      0.02        e20    0.0199   e20
                                                      0.16       101           -    -
                                                      0.05     1 427           -    -
          Rosser & Schoenfeld (1962) [42]
                                                      0.02     7 481           -    -
                                                         -          -   0.01624     1
                                                     0.015    11 927           -    -
                                                     0.010    32 057           -    -
          Rosser & Schoenfeld (1975) [43]            0.005    89 387           -    -
                                                     0.002   487 381           -    -
                                                  0.001316 1 319 007           -    -
                                                         -          - 0.001102      1
                                                  0.001303 1 155 901           -    -
          Schoenfeld (1976) [46]                         -          - 0.001093      1
          Platt & Trudgian (2016) [36]                   -          - 7.5 · 10−7    0


  In the case k = 0 and X0 = 1 of Theorem 1 we are able to reduce the bound 1.75 · 10−7 due to
Platt and Trudgian ([36], see Table 1) to 1.94 · 10−8 :

Corollary 2.1. We have

                            θ(x) ≤ (1 + 1.93378 · 10−8 )x       for all x ≥ 0

Proof. We combine together the fact that θ(x) < x for 0 < x ≤ 1019 ([4, Theorem 2, (1.7)]) and
that M0 = ε(19 log 10) = 1.93378 · 10−8 (from Table 8 with k = 0 and X1 = 1019 ).            



                                           Table 2. Case k = 1

       Author                                          m1        X0              M1        X1
                                                      2.85         2            2.85         2
       Rosser (1941) [40]
                                                      0.96     e2000            0.96     e2000
                                                      0.50       563            0.50         1
       Rosser & Schoenfeld (1962) [42]
                                                      0.47       569            0.47       569
                                                 0.02500     678 407       0.02500     678 407
       Rosser & Schoenfeld (1975) [43]
                                                 0.02424     758 699       0.02424     525 752
       Schoenfeld (1976) [46]                    0.02400     758 711       0.02400     758 711
       Dusart (1999) [12]                    6.788 · 10−3 10 544 111   6.788 · 10−3 10 544 111
       Dusart (2010) [13, Table 6.4-6.5]     3.888 · 10−5        e35   3.888 · 10−5        e35




In the case k = 1 and X0 = e35 ≈ 1.586 · 1015 of Theorem 1, we reduce the bound 3.888 · 10−5 due
to Dusart ([13], see Table 2) to 1.0778 · 10−6 .
                                                     4
                                        Table 3. Case k = 2

        Author                                       m2         X0             M2         X1
        Rosser and Schoenfeld (1975) [43]        8.6853           1        8.6853           1
        Schoenfeld (1976) [46]                   8.0720           1        8.0720           1
        Dusart (1999) [12]                            0.2 3 594 641             0.2 3 594 641
        Dusart (2010) [13, Table 6.4-6.5]   0.140 · 10−3        e35   0.140 · 10−3        e35
                                                       −3                        −3
        Trudgian (2016) [50, Lemma 1]       0.450 · 10          e35   0.450 · 10          e35



Note that in the case k = 2 and X0 = e35 , Dusart’s calculation [13] of 0.140 · 10−3 was based
on assuming Wedenevski-Gourdon’s verification of the Riemann Hypothesis up to height 1013 [52]
while Trudgian’s [50] 0.450 · 10−3 is “worse” as based on Platt’s rigorous verification at the lower
height of 3 · 1010 [34]. We improve both of these results by obtaining m2 = M2 = 5.9771 · 10−5 for
X0 = X1 = e35 .

                                        Table 4. Case k = 3

                   Author                                  m3 X 0        M 3 X1
                   Rosser and Schoenfeld (1975) [43]    11 762   1    11 762   1
                   Schoenfeld (1976) [46]               10 644   1    10 644   1
                   Dusart (1999) [12]                      515   1       515   1
                   Dusart (2010) [13, Table 6.4-6.5]      0.35 e30      0.35 e30



In the case k = 3 and X0 = X1 = e30 of Theorem 1, we obtain m3 = M3 = 0.0244. Observe that
this improves Dusart’s [13] bound of 0.35 (see Table 4 below). In addition, we recover Axler’s [1,
Theorem 1.1]: for x ≥ 19 035 709 163 > e23 , m3 = 0.15 and for x ≥ 1, M3 = 0.15. In particular,
0.15 is attained at the prime p841 508 302 = 19 035 709 163.

                                        Table 5. Case k = 4

               Author                                        m4 X 0            M 4 X1
               Rosser and Schoenfeld (1975) [43]    1.8559 · 107 1    1.8559 · 107  1
               Schoenfeld (1976) [46]                16 570 000  1     16 570 000   1
               Dusart (1999) [12]                      1 717 433 1       1 717 433  1
               Dusart (2010) [13, Tables 6.4-6.5]          1 300 1           1 300  1



In the case k = 4 and X0 = X1 = 1 of Theorem 1, we have m4 = M4 = 151.3: as noticed by Dusart
[15], the value 151.2235 . . . is a max attained at the prime 1 423. For X0 = X1 = 7 · 1011 , we find
m4 = M4 = 57.184.
1.3. The conjectural size of θ(x). In this article, we have attempted to establish the best-known
Chebyshev-type bounds for θ(x). Note that de la Vallée Poussin’s proof of the prime number
                                                            1
theorem [51] actually yields θ(x) = x + O(x exp(−c(log x) 2 )) for some c > 0. Furthermore, Platt
and Trudgian [37] have established for x0 ≥ 1000, there exist positive constants A, B, C such that
for all x ≥ ex0
                                                                       
                                                                   x 12
(1.4)                        |θ(x) − x| ≤ A( logR x )B exp −C( log
                                                                 R  )     .
                                                    5
Recently, Büthe [3] has shown that under partial RH (true when 1 ≤ |γ| ≤ T ), then
                                                  1 √
(1.5)                               |θ(x) − x| ≤      x(log x)2
                                                 8π
                                            1
for all x ≥ 599 satisfying 4.92( logx x ) 2 ≤ T . For instance, for Platt’s T = 3.061 · 1010 , (1.5) holds for
599 ≤ x ≤ 1.89 · 1021 . Conditionally on RH being true, Schoenfeld [46, Theorem 10] has shown that
(1.5) holds for all x ≥ 599. These are effective versions of a theorem of von Koch [26]. To date, the
           1
constant 8π  has not been improved. It may be asked, what is the true size of the error term on the
right hand side of (1.5). The explicit formula of Riemann tells us that on the Riemann hypothesis
                               θ(x) − x                    X xiγ 
                                   √         = −1 − 2Re         1       + ···
                                       x                          + iγ
                                                            γ>0 2

where 21 +iγ ranges through the non-trivial zeros of zeta. It is known under the Linear Independence
                                                                     √
Hypothesis (LI) 2 that the distribution of values of (θ(x) − x)/ x is the same as that of the
                           P         Xγ   
random variable X = 2Re       γ>0 | 1 +iγ| where the Xγ are independent random variables, uniformly
                                        2
distributed on the unit circle.
                        √ √ By giving sharp estimates for the
                                                          √ probability
                                                              √         of the tail of X, it may
be shown that exp(−c2 V e 2πV ) ≤ P (x ≥ V ) ≤ exp(−c1 V e 2πV ), for some c1 , c2 > 0. This
suggests that
                                 θ(x) − x        1                θ(x) − x        1
                    lim sup √               2
                                              =    and lim inf √             2
                                                                               =− .
                     x→∞        x(log log x)    2π      x→∞      x(log log x)    2π
This conjecture in the case of ψ(x) is due to Montgomery [30].

                                            2. Bounding ψ(x) − θ(x).
  In this section we give bounds for ψ(x) − θ(x) of the shape
                                                           1            1
(2.1)                            ψ(x) − θ(x) ≤ a1 x 2 + a2 x 3 for all x ≥ x0
where a1 and a2 depend on x0 . Rosser and Schoenfeld [43, Theorem 6] established this with
a1 = 1.001102, a2 = 3, and x0 = 1. Recently, Dusart [15, Corollary 4.5] improved this bound to
a1 = 1 + 1.47 · 10−7 and a2 = 1.78 for x0 = 1. As we apply such a bound to various other values of
x0 , we are able to reduce the values of a1 and a2 .
Proposition 3. Let x0 ≥ 29 . Let α > 0 exist such that θ(x) ≤ (1 + α)x for x > 0. Then for
x ≥ x0 ,
                                     log x 
                                      X
                                      log 2
                                                  1        1
(2.2)                                         θ(x k ) ≤ ηx 3
                                                  k=3

where
                                                                     log x 
                                                                           0   
(2.3)                         η = η(x0 ) = (1 + α) max f (x0 ), f (2 log 2 +1 )
with
                                                            log x 
                                                             log 2
                                                             X          1   1
(2.4)                                           f (x) :=               xk−3 .
                                                               k=3

  2LI is the conjecture that the positive ordinates of the zeros of ζ(s) are linearly independent over Q.
                                                               6
                                                            1                                               1
Proof. Let x ≥ x0 ≥ 29 . Bounding each θ(x k ) term of (2.2) by (1 + α)x k yields
                                                                   log x 
                                                    1               log 2
                             ψ(x) − θ(x) − θ(x 2 )                  X        1   1
                                        1             ≤ (1 + α)             xk−3 .
                                      x3                            k=3
                                                                                         log x 
                                                                                               0 +1
Next, we divide the interval [x0 , ∞) as follows. If x0 is not a power of 2, then 2 log 2           is the least
power of 2 in [x0 , ∞). Thus, we have
                                            log x            [∞
                                                  0
                         [x0 , ∞) = [x0 , 2 log 2 +1 ) ∪                  [2n , 2n+1 ).
                                                             log x 
                                                                            n=           0   +1
                                                                                    log 2


Observe that f (x) decreases on [2n , 2n+1 ) and thus f (x) ≤ f (2n ) for every x ∈ [2n , 2n+1 ). Note
                                P        n n
that f (2n ) = 1 + un where un = nk=4 2 k − 3 . We now show that un+1 < un for n ≥ 9. We have
                           n
                           X      n+1    n+1            1       1               n+1              n+1
                                                                                                          n
                                                                                                           X  n+1   1   1
                                                                                                                              
(2.5)       un+1 − un =          2 k − 3 (1 − 2 3 − k ) + 21− 3 = 2− 3                                  2−   2 k (2 3 − k − 1) .
                           k=4                                                                              k=4

Observe that if n ≥ 20, then
                        n
                        X      n+1   1    1                 n+1         1   1                    21     1
                              2 k (2 3 − k − 1) > 2 4 (2 3 − 4 − 1) ≥ 2 4 (2 12 − 1) > 2
                        k=4
and it follows that un+1 − un < 0 for n ≥ 20. Finally, a numerical calculation verifies that the
right hand side of (2.5) is negative for 9 ≤ n ≤ 19.
                                                     Therefore
                                                               it follows that f (2n ) > f (2n+1 )
                                               log x0                       log x0
                                                     +1
for n ≥ 9. Therefore     log xf (x)
                                  ≤ f (2
                                               log 2    ) on [2 log 2  +1 , ∞).   Similarly,
                                                                                 log           we see that f (x) ≤
                               0 +1                                     log x0         x0
                                                                                            +1
f (x0 ) for x ∈ [x0 , 2 log 2      ), since f (x) decreases on [2 log 2 , 2 log 2            ). In summary, f (x) ≤
                   log x0
                            +1   
max f (x0 ), f (2 log 2         ) .                                                                                 

   We now apply the case where x0 = eb to obtain the following corollary.
Corollary 3.1. Let b ≥ 7. Assume x ≥ eb . Then we have
                                                                                1            1
                                              ψ(x) − θ(x) − θ(x 2 ) ≤ ηx 3
where
                                                                               
                                                                            b
                                                                                 +1
(2.6)                         η = (1 + 1.93378 · 10−8 ) max f (eb ), f (2 log 2     )

and f is defined by (2.4).
Proof. We apply Proposition 3 with α = 1.93378 · 10−8 from Corollary 2.1 and x0 = eb where we
observe that x0 = eb ≥ e7 > 29 .                                                            
  The next result is a general version of [43, Theorem 6, eq (5.3)] and [15, Corollary 4.5]. The two
inputs we take are a Chebyshev Bias constant x1 such that θ(x) < x for x ≤ x1 and a bound for
|ψ(x) − x| for x ≥ y0 for every y0 > 0. These are used in conjunction with Corollary 3.1 to prove
Theorem 5.
Proposition 4. Let b ≥ 7 and assume that for fixed b, there exists a positive constant ε(b) such
that
(2.7)                                    |ψ(x) − x| ≤ ε(b)x for all x ≥ eb .
                                                                    7
Assume there exists x1 ≥ e7 such that
                                                  θ(x) < x for all x ≤ x1 .
If b ≤ 2 log x1 , then we have
                                      1                                      1
(2.8)                             θ(x 2 ) < (1 + ε(log x1 ))x 2 for x ≥ eb .
If b > 2 log x1 , then we have
                                          1                              1
                                   θ(x 2 ) < (1 + ε(b/2))x 2 for x ≥ eb .
                       1
Proof. We bound θ(x 2 ) by cases depending on the range of x we are considering.
                                            1
Case 1: eb ≤ x21 . If eb ≤ x ≤ x21 , then x 2 ≤ x1 , and thus
                                                   1         1
                                          θ(x 2 ) < x 2 for eb ≤ x ≤ x21 .
                           1
On the other hand, if x 2 > x1 = elog x1 , then we have by (2.7)
                                          1              1                                1
                                   θ(x 2 ) ≤ ψ(x 2 ) ≤ (1 + ε(log x1 ))x 2 ,
                                                                     1
since log x1 ≥ 7. The last two inequalities for θ(x 2 ) combine to establish (2.8).
Case 2: eb > x21 . As in the above subcase, we have for x ≥ eb
                                              1              1                        1
                                    θ(x 2 ) ≤ ψ(x 2 ) ≤ (1 + ε(b/2))x 2 ,
        1    b
since x 2 > e 2 > x1 ≥ e7 .                                                                              
   Using the previous general results we obtain the following explicit bounds for ψ(x) − θ(x) of the
shape (2.1). This generalizes Rosser and Schoenfeld’s [43, Theorem 6] and Dusart’s [15, Corollary
4.5] results.
Theorem 5. Let α > 0 exist such that
                                      θ(x) ≤ (1 + α)x for all x > 0.
Assume for every b ≥ 7 there exists a positive constant ε(b) such that
                                     ψ(x) − x ≤ ε(b)x for all x ≥ eb .
Assume there exists x1 ≥ e7 such that
(2.9)                                              θ(x) < x for x ≤ x1 .
Let b ≥ 7. Then, for all x ≥ eb we have
                                                                         1        1
                                          ψ(x) − θ(x) < a1 x 2 + a2 x 3 ,
where                                 (
                                       1 + ε(log x1 )                    if b ≤ 2 log x1 ,
                                 a1 =
                                       1 + ε(b/2)                        if b > 2 log x1 ,
and                                                                        
                                                                   b
                                 a2 = (1 + α) max f (eb ), f (2⌊ log 2 ⌋+1 ) .

                                    1    P⌊ log  x
                                                   ⌋     1
Proof. We have ψ(x)− θ(x) = θ(x 2 ) + k=3  log 2
                                                     θ(x k ). For any b ≥ 7, setting x0 = eb in Proposition
              P log   x
                  log 2     1         1                                           1
4, we bound k=3         θ(x k ) by ηx 3 as defined in (2.3). We bound θ(x 2 ) with Proposition 4 by
taking either a1 = 1 + ε(log x1 ) for b ≤ 2 log x1 or a1 = 1 + ε(b/2) for b > 2 log x1 .                  
  Using the best known values for x1 , α and ε(b), we have the following Corollary:
                                                                 8
Corollary 5.1. Let b ≥ 7. Then for all x ≥ eb we have
                                                             1        1
(2.10)                                ψ(x) − θ(x) < a1 x 2 + a2 x 3 ,
where
                                       (
                                        1 + 1.93378 · 10−8         if b ≤ 38 log 10,
(2.11)                   a1 = a1 (b) =
                                        1 + ε(b/2)                 if b > 38 log 10,
                                                                             b
                                                                                       
(2.12)               a2 = a2 (b) = (1 + 1.93378 · 10−8 ) max f (eb ), f (2⌊ log 2 ⌋+1 ) ,
where f is defined by (2.4) and values for ε(b/2) are from Table 8.
Proof. We apply Theorem 5 with x0 = eb and       α = 1.93378    · 10−8 from Corollary 2.1. Thus we
                                                  b
                                                           
get a2 = (1 + 1.15177 · 10−8 ) max f (eb ), f (2 log 2 +1 ) . In Proposition 4, we take x1 = 1019 from
the work of Büthe [4, Equation (1.7)], and get ε(log x1 ) = 1.93378 · 10−8 from Table 8. Thus for
b ≤ 38 log 10 we have a1 = 1 + 1.93378 · 10−8 and for b > 38 log 10 we take ε(b/2) from Table 8.
In the case that b/2 is not included in the table, we bound b/2 by the greatest value smaller than
b/2.                                                                                             
  We have the following values for a2 .
               b     20      25         30   35        40       43       50
              a2 1.4263 1.2196 1.1211 1.07086 1.04320 1.03253 1.01718
    b       100                150            200              250              300
   a2 1 + 2.421 · 10−4 1 + 3.749 · 10−6 1 + 7.712 · 10−8 1 + 2.024 · 10−8 1 + 1.936 · 10−8

                           3. Bounds for θ(x) − x of the form (logxx)k

In this section we prove Theorem 1, our main theorem for giving estimates for θ(x)−x of size (logxx)k .
More precisely, we prove that for any k = 0, . . . , 5 and any X0 , X1 ≥ 1, there exists mk , Mk > 0
such that
                                              
                                         mk
(3.1)                          x 1−              ≤ θ(x)     for all x ≥ X0
                                      (log x)k
and
                                                     
                                                Mk
(3.2)                         θ(x) ≤ x 1 +                  for all x ≥ X1 .
                                             (log x)k
The values for X0 , X1 , mk , and Mk may be found in Table 14 for k = 0, and in Table 15 for
k = 1, . . . , 5. Theorem 1 is a generalization of Axler’s [1, Theorem 1]. We separate the cases
k = 0 and k = 1, . . . , 5 (with k = 0 being treated at the end of this section). For k = 1, . . . , 5, we
subdivide the interval [1, ∞) (for the range of x) as follows:
                           [1, ∞) = [1, eJ0 ) ∪ [eJ0 , eJ ) ∪ [eJ , eK ) ∪ [eK , ∞).
We now explain how the values of J0 , J and K are chosen. For shorthand, we respectively call
[1, eJ ), [eJ , eK ), and [eK , ∞) the “small”, “middle”, and “large” ranges of x.
                                                                                               √      
       • In the large range of x, we apply bounds for θ(x) of the shape x(log x)c exp −C log x .
       • In the middle range of x, we subdivide the interval [eJ , eK ) into smaller consecutive intervals
                  ′
          [eb , eb ). In each such subinterval, we make use of bounds for ψ(x) of the shape |ψ(x) − x| ≤
          εx and for the difference ψ(x) − θ(x) (established respectively in Appendix A and Section
          2). By numerical experimentation we choose
                                                K = 25 000.
                                                      9
        • In the small range of x, upper bounds for θ(x) are the result of direct calculations, namely
          that θ(x) < x is known for all x < eJ . Here Büthe’s [4, Theorem 2] allows us to take
                                         J = 19 log 10 = 43.74 . . . .
        • In the small range of x, we subdivide [1, eJ ) at eJ0 to obtain lower bounds for θ(x):
            – We do direct calculations up to eJ0 where we use
                                             J0 = log(7 · 1011 ).
            – For x ∈ [eJ0 , eJ ) we use a little known comparison of ψ(x)√with θ(x) due to Costa
              Pereira [6], together with numerical bounds for (ψ(x) − x)/ x, computed by Büthe
              [4].
3.1. Upper and lower bounds for θ(x) in the large range x ≥ eK . The following lemma
derives a bound of the form x/(log x)k for θ(x)−x from the classical de la Vallée Poussin-Hadamard
bound.
Lemma 6. Suppose there exists c1 , c2 , c3 , c4 > 0 such that
                                                                          1
(3.3)                   |θ(x) − x| ≤ c1 x(log x)c2 exp(−c3 (log x) 2 ) for all x ≥ c4 .
                                                   2
Let k > 0 and let b ≥ max(log c4 , log( 4(c2c+k)
                                             2   )). Then for all x ≥ eb we have
                                              3

                                                             Ak (b)x
(3.4)                                      |θ(x) − x| ≤               ,
                                                             (log x)k
where
                                                                     √
(3.5)                                    Ak (b) = c1 · bc2 +k e−c3 b .
                                                             1                   c1 g(x)x
Proof. We denote g(x) = (log x)c2 +k exp(−c3 (log x) 2 ). By (3.3), |θ(x) − x| < (log x)k
                                                                                          for all x ≥ c4 .
                                                                                      2
It suffices to bound g: by calculus, g(x) decreases when x ≥ 4(c2c+k)
                                                                  2   . Therefore |θ(x) − x| ≤
                                                                                  3
c1 g(eb )x
 (log x)k
           . Note that c1 g(eb ) = Ak (b) and the condition on b follows from the conditions eb ≥ c4 and
               2
eb ≥ 4(c2c+k)2   .
             3
                                                                                                        
  The current best explicit version of (3.3) is due to Platt and Trudgian. Details are given in
Corollary 14.1 in Appendix A.
Theorem 7. [37, Theorem 1] Let x0 ≥ 1000 and let R be a formal constant such that there exists
                                                 1
a zero-free region of the form Re(s) ≥ 1 − R log |Im(s)| for |Im(s)| ≥ 2. There exist positive constants
                                 x
(A, B, C) such that for all x ≥ e 0
                                                                              1
                                |θ(x) − x| < Ax( logR x )B exp(−C( logR x ) 2 ).
  Using this we obtain the following corollary.
Corollary 7.1. Let k > 0, x0 ≥ 1000, and b ≥ max(log x0 , log(4R( B+k 2
                                                                   C ) )). Then
                                                  Ak (b)x
                                 |θ(x) − x| ≤                    for all x ≥ eb
                                                  (log x)k
where
                                                                          r !
                                            A                              b
(3.6)                              Ak (b) = B · bB+k · exp −C
                                           R                               R
and R = 5.573412. Values for Ak (b) for 1 ≤ k ≤ 5 are displayed in Table 9.
                                                       10
Proof. We apply Lemma 6 with R = 5.573412 (using [31]), and values from Theorem 7, namely
c1 = RAB , c2 = B, c3 = √CR , and c4 = x0 . We complete the proof by noticing

                                     4(c2 + k)2       B + k 2
                                                = 4R            .
                                         c23             C
                                                                                                              

3.2. Upper and lower bounds for θ(x) in the middle range eJ ≤ x < eK . In this range we
combine bounds for ψ(x) − θ(x) established in Corollary 5.1 with the current best known bounds
for ψ(x) as derived in Appendix A to produce a bound for θ(x). We begin with a general result.
Lemma 8. Let k = 1, . . . , 5. Assume there exist a positive integer n, real numbers aℓ ≥ 0 for every
ℓ ∈ {1, 2, . . . , n}, and x0 > 0 such that
                                              n
                                              X             1
(3.7)                       ψ(x) − θ(x) ≤            aℓ x ℓ+1          for all x ≥ x0 .
                                              ℓ=1

Let b′ > b ≥ 2k, eb ≤ x0 , and assume that there exists ε(b) > 0 such that
(3.8)                           |ψ(x) − x| ≤ ε(b)x                for all x ≥ eb .
Then we have
                                              Bk x                                       ′
(3.9)                       |θ(x) − x| ≤                         for all x ∈ [eb , eb ]
                                            (log x)k
where
                                                    n
                                                    X                      ℓ
                                                                                            
(3.10)             Bk = Bk (b, b′ ) = max ′                aℓ (log x)k x− ℓ+1 + ε(b)(log x)k .
                                     x∈[eb ,eb ]
                                                     ℓ=1
Note that
                                                     n
                                                     X                         
                          ek = B
                               ek (b, b′ ) = bk                       ℓb
(3.11)               Bk ≤ B                                 aℓ exp −                + ε(b)(b′ )k .
                                                                     ℓ+1
                                                     ℓ=1

Remark. Note that the value for B is slightly smaller than B̃. However, at times we make use of
the weaker value given by (3.11).
Proof. By the triangle inequality and the non-negativity of ψ(x) − θ(x), we have
                               |θ(x) − x| ≤ ψ(x) − θ(x) + |ψ(x) − x|.
Bounding these terms by (3.7) and (3.8), we have for x ≥ eb ,
                                   x X                                        
                                          n
                                                             ℓ
                                                        k − ℓ+1              k
                   |θ(x) − x| ≤              a ℓ (log x) x      + ε(b)(log x)    .
                                (log x)k
                                                   ℓ=1
                                                                                                     k(ℓ+1)
This immediately implies (3.9) holds with (3.10). Observe that since x ≥ eb > e2k ≥ e ℓ , then
                     ℓ
each aℓ (log x)k x− ℓ+1 decreases with x. On the other hand, ε(b)(log x)k increases with x and thus
we have the inequality (3.11).                                                                   
Corollary 8.1. Let k ∈ {1, . . . , 5}, and let b and b′ be any consecutive entries of column 1 of Table
8 such that b < b′ . i.e. we assume that there exists ε(b) > 0 such that
                                                                                     ′
                             |ψ(x) − x| ≤ ε(b)x                 for all x ∈ [eb , eb ].
                                                           11
Thus
                                               Bk (b, b′ )x                             ′
(3.12)                         |θ(x) − x| ≤                         for all x ∈ [eb , eb ],
                                                (log x)k
where
                                                       b                   2b
(3.13)                        Bk (b, b′ ) = a1 (b)bk e− 2 + a2 (b)bk e− 3 + (b′ )k ε(b),
and a1 , a2 are defined in Corollary 5.1.
In addition, let b0 be any entry in column 1 of Table 10. Then,
                                                Bk (b0 )x
(3.14)                          |θ(x) − x| ≤                       for all x ∈ [eb0 , eK ]
                                                (log x)k
where K is the last entry in Column 1 of Table 10, and
(3.15)                                       Bk (b0 ) = max
                                                          ′
                                                            Bk (b, b′ ).
                                                           b,b
                                                        b0 ≤b<b′


Values for Bk (b, b′ ) and Bk (b0 ) are respectively displayed in Tables 10 and 11.
Proof. We apply Lemma 8 with k ∈ {1, 2, 3, 4, 5}, b0 = b, and n = 2 and obtain (3.13). For we take
              ek (b, b′ , 2) with a1 = a1 (b) and a2 = a2 (b) as defined in (2.11) and (2.12) respectively.
Bk (b, b′ ) = B
                                                                                      S               ′
The inequality (3.14) follows from (3.12) together with the fact that [eb0 , eK ] = b∈[b0 ,K) [eb , eb ]. 

Remark 1. Note that Bk (b, b′ ) is essentially ε(b)(b′ )k and thus any refinement on the other terms only
brings minor improvements. For instance, Costa-Pereira’s [6, Theorem 1] estimates for ψ(x) − θ(x)
affects digits much further than those we display. For every x > 0,
                          1            1          1                               1           1   1
(3.16)       ψ(x) − ψ(x 2 ) − ψ(x 3 ) − ψ(x 5 ) ≤ θ(x) ≤ ψ(x) − ψ(x 2 ) − ψ(x 3 ) − ψ(x 7 ).
2. While we use the bound
                                                       b                   2b
                              Bk (b, b′ ) ≤ a1 (b)bk e− 2 + a2 (b)bk e− 3 + ε(b)(b′ )k ,
for any given b and k, a more specific max in (3.13) can be computed exactly using calculus.

3.3. Upper bounds for θ(x) in the small range x < eJ . It is has been proven that the first
sign change of x − θ(x) occurs before 1.3972 · 10316 [45, Lemma 9.4]. In fact, Büthe [4, Theorem 2,
(1.7)] has shown that
                                                √
(3.17)                          θ(x) < x − 0.05 x for all x ≤ 1019 .
It follows that
(3.18)            θ(x) − x ≤ Mk for all x ≤ eJ0 , with Mk = 0 and J = 19 log 10.

3.4. Lower bounds for θ(x) in the small range eJ0 ≤ x < eJ . We provide here an improvement
to Büthe’s [4, Lemma 1].
                                         x − ψ(x)
                                   −c ≤     √     ≤ C.
                                              x
Lemma 9. Let 1 ≤ u < v. Assume there exist c = cu,v > 0 and C = Cu,v > 0 such that
                                           x − ψ(x)
(3.19)                          −c≤           √     ≤C             for every x ∈ [u, v].
                                               x
Assume that there exists c0 > 0 such that
(3.20)                                       ψ(x) < c0 x for all x > 0.
                                                            12
If u2 < v, then
                                                       1              1                    1                    1
(3.21)              θ(x) ≥ x − (C + 1)x 2 − c0 x 3 − cx 4 − c0 x 5 for all x ∈ [u2 , v].
Proof. Costa-Pereira [6, Theorem 1, Equation (1)] proved that
                                                               1                   1                        1
                         ψ(x) − θ(x) ≤ ψ(x 2 ) + ψ(x 3 ) + ψ(x 5 ) for all x > 0.
Together with (3.20), it follows
                                                               1               1                        1
                         ψ(x) − θ(x) ≤ ψ(x 2 ) + c0 x 3 + c0 x 5 for all x ∈ [u, v].
This may be rewritten as
                                                                                                  1                   1
                                                                                                                                !
                                   1       ψ(x) − x                   1                1           x 2 − ψ(x 2 )                           1        1
(3.22)          θ(x) ≥ x + x       2
                                                 1             −x +x  2                4
                                                                                                                    1               − c0 x 3 − c0 x 5 .
                                             x   2                                                          x       4

                                                                          1                1
We conclude using (3.19): ψ(x)−x ≥ −C and x −ψ(x )                        2                2
                             1                1    ≥ −c for every x ∈ [u2 , v].                                                                                 
                                       x2                                     x4

  Rosser and Schoenfeld [42, Theorem 12] proved (3.20) with
(3.23)                                                             c0 = 1.03883.
Note that this value cannot be improved since the bound (3.20) is achieved for x = 113. We shall
use this value throughout the article. Recently, Büthe has proven many bounds like (3.19). From
[4, Equation (6.2), Table 1], we have:
                                                      u      v       c   C
                                                     100 5 · 1010 0.8 0.81
                                                     100 32 · 1012 0.88 0.86
                                                     100   1019    0.94 0.94
In the following corollary, we restrict x to [eb , v], a subset of [u2 , v].
Corollary 9.1. Let (v, c, C) ∈ {(5 · 1010 , 0.8, 0.81), (32 · 1012 , 0.88, 0.86), (1019 , 0.94, 0.94)}. Let
k ≥ 0 and let b satisfy max(104 , e2k ) ≤ eb ≤ v. Then
                                                             Cb,k x
(3.24)                                 θ(x) ≥ x −                                          for all x ∈ [eb , v]
                                                           (log x)k
where
(3.25)                  Cb,k = bk ((C + 1)e−b/2 + c0 e−2b/3 + ce−3b/4 + c0 e−4b/5 ),
and where c0 is defined in (3.23). Values of Cb,k can be found in Table 12.
                                              b
Proof. We apply (3.21) with u = e 2 :
                                                       1              1                        1                1
                     θ(x) ≥ x − (C + 1)x 2 − cx 4 − c0 x 3 − c0 x 5 for all x ∈ [eb , v].
We now set
                                   n                 (log x)k                  (log x)k                             (log x)k               (log x)k o
(3.26)          Cb,k = max             (C + 1)             1         + c0                      2        +c                  3       + c0        4       .
                       x∈[eb ,v]                       x2                              x3                               x4                     x5
                                                                                                                                                            k
We find that this equals the expression in (3.25) by observing that for a ∈ { 21 , 23 , 34 , 45 }, (logxax) is
decreasing for x ≥ eb as long as eb ≥ ek/a . This last inequality leads to the condition b ≥ 2k.            
                                                                              13
3.5. Lower bounds for θ(x) for x < eJ0 . The following lemma gives a condition to obtain a
lower bound for θ(x) for the first values of x.
Lemma 10. Let k = 1, . . . , 5. Let 0 < a < b such that a > ek+1 . Let pn denote the n-th prime,
with pn0 and pn1 being the smallest primes greater than a and b respectively. Let
                                                   (log pn )k · (pn − θ(pn−1 ))
(3.27)                      Dk (a, b) =     max                                 .
                                          n0 ≤n≤n1               pn
If
(3.28)                                     Dk (a, b) < (k + 1)k+1 ,
then
                                              Dk (a, b)x
(3.29)                          θ(x) ≥ x −               when a ≤ x ≤ b.
                                              (log x)k
Proof. Fix n0 ≤ n ≤ n1 and set
                                               (log pn )k (pn − θ(pn−1 ))
(3.30)                             Dk (n) =                               .
                                                            pn
Let x ∈ [pn−1 , pn ) and observe that
                                                                              
                                                                    Dk (n)
                                θ(x) = θ(pn−1 ) = pn           1−                  ,
                                                                  (log pn )k
and that the function ϕ(x) = x(1 − c/(log x)k ) increases with x, as long as c < (k + 1)k+1 . This is
                        c                      ′′          ck                                         ′
since ϕ′ (x) = 1 + (log x)k+1 (k − log x) and ϕ (x) = x(log x)k+2 (log x − (k + 1)). It follows that ϕ has

a minimum at x = ek+1 . Therefore ϕ is increasing since
                                                                   c
                               ϕ′ (x) ≥ ϕ′ (ek+1 ) = 1 −                  > 0,
                                                               (k + 1)k+1
from the assumption c < (k + 1)k+1 . Thus (3.28) ensures that
                                                    
                                             Dk (n)
(3.31)                      θ(x) ≥ x 1 −               for all x ∈ [pn−1 , pn ).
                                            (log x)k
                                           S
It follows that (3.29) holds since [a, b] ⊂ n0 ≤n≤n1 [pn−1 , pn ) and Dk (a, b) = maxn0 ≤n≤n1 Dk (n). 
Methodology: We use this lemma to obtain a numerical lower bound for θ(x) on [1, 7.0 × 1011 ].
We first subdivide into the intervals In = [(n − 1) · 1010 , n · 1010 ] with n ranging from 1 to 70. We
subdivide each In further into 100 subintervals, each of length 108 . We apply the Lemma 10 to
each of these subintervals, recording the corresponding Dk value. For each n, we take for Dk (n) the
largest value among all Dk ’s arising from the 100 subintervals. Values for Dk for selected ranges in
[1, 7.0 × 1011 ] are recorded in Table 13.
3.6. The case k = 0. The upper bound is a direct result of Theorem 2 and the partial verification
of θ(x) < x.
Lemma 11. Let b > 0. Assume
(3.32)                                    θ(x) < x for all x ≤ eb ,
and that there exists ε(b) > 0 such that
(3.33)                            |ψ(x) − x| ≤ ε(b)x for all x ≥ eb .
Then we have
(3.34)                            θ(x) ≤ (1 + ε(b))x, for all x > 0.
                                                      14
In addition
                                      b     2b     4b
                                                        
(3.35)               1 − ε(b) − c0 (e− 2 + e− 3 + e− 5 ) x ≤ θ(x), for all x ≥ eb .

Proof. Inequality (3.34) follows immediately from Theorem 2 and the fact that θ(x) ≤ ψ(x). Now
                                                                     1        1       1
for every x > 0, [6, Theorem 1] asserts that ψ(x) − θ(x) ≤ ψ(x 2 ) + ψ(x 3 ) + ψ(x 5 ). Together
                                  1     1     1
with (3.20), we have ψ(x) − c0 (x 2 + x 3 + x 5 ) ≤ θ(x), and we conclude by applying Theorem 2 to
ψ(x).                                                                                           

3.7. Proof of Theorem 1.
If k = 0, we apply Lemma 11 with b = log X1 for the upper bound and b = log X0 for the lower
bound. We can define:
                                              −1    −2      −4
                   m0 = ε(log X0 ) + c0 (X0 2 + X0 3 + X0 5 ) and M0 = ε(log X1 ).
For the rest of this section, we assume k ∈ {1, . . . , 5}. Throughout this proof we let bn denote the
n-th entry of column 1 of Table 8. Furthermore, let K = 25000 be the largest b value in Table 8,
and let
(3.36)                                    X0 = eu0 and X1 = eu1 .
The proof consists of four cases. We shall divide up the intervals [Xi , ∞) for i = 0, 1 into various
subintervals. On each of the subintervals we shall apply a combination of Corollary 7.1, Corollary
8.1, Corollary 9.1, and Lemma 10. By combining together our various bounds we shall establish
the required bounds (3.1) and (3.2). If X0 , X1 ≥ eK , then only Corollary 7.1 is considered.

3.7.1. Case 1: X0 , X1 ≥ eK . By Corollary 7.1, we can take
                                   mk = Ak (u0 ) and Mk = Ak (u1 ).
For the second case, we consider eJ ≤ Xi ≤ eK . Here we consider Corollary 7.1 and Corollary 8.1.
For all following cases, let Bi,k = Bk (b, b′ ) where b is the i-th entry in column 1 of Table 10. Validity
for all x ≥ X0 or x ≥ X1 can be established by applying Corollary 8.1 over consecutive intervals,
then applying Corollary 7.1 to bound for all x above the end of the intervals. Thus we now have
the following case:

3.7.2. Case 2: eJ ≤ X0 , X1 < eK . Combining Corollary 7.1 and Corollary 8.1, we may take
              mk = max(Ak (bℓ+1 ), max (Bi,k )) and Mk = max(Ak (bℓ+1 ), max (Bi,k ))
                                    n0 ≤i≤ℓ                                    n1 ≤i≤ℓ

where n0 and n1 are the greatest natural numbers such that bn0 ≤ u0 and bn1 ≤ u1 and ℓ ≥ n0 or
ℓ ≥ n1 is chosen to minimize the values of mk and Mk .
Remark. Note that whenever possible, the value of ℓ is chosen such that Ak (bℓ+1 ) ≤ maxn0 ≤i≤ℓ (Bi,k ).
  For the last two cases, we denote j ∗ the row of Table 10 where bj ∗ = J = 19 log 10.

3.7.3. Case 3: eJ0 ≤ Xi ≤ eJ . From Corollary 7.1, Corollary 8.1, (3.18), and Corollary 9.1, we can
take
          mk = max(Ak (bℓ+1 ), max (Bi,k ), C⌊u0 ⌋,k ) and Mk = max(Ak (bℓ+1 ), max (Bi,k ))
                                 j∗≤i≤ℓ                                            j∗≤i≤ℓ

where ℓ ≥ n0 or ℓ ≥ n1 is chosen to minimize the values of mk and Mk .
                                                    15
3.7.4. Case 4: Xi < eJ0 . By applying Lemma 10 we obtain
                                             
                               Dk (X0 , eJ0 )
                         x 1−                   ≤ θ(x) when X0 ≤ x ≤ eJ0 .
                                  (log x)k
We combine this with Corollary 7.1, Lemma 8, Corollary 9.1, and (3.18):
   mk = max(Ak (bℓ+1 ), max (Bi,k ), CJ0 ,k , Dk (X0 , eJ0 )), and Mk = max(Ak (bℓ+1 ), max (Bi,k ))
                          j∗≤i≤ℓ                                                                  j∗≤i≤ℓ

where ℓ ≥ n0 or ℓ ≥ n1 is chosen to minimize the values of mk and Mk .

3.8. Computational examples. We now give examples of how to apply Theorem 1 in specific
cases. First, we are able to improve the second part of [1, Theorem 1].
Corollary 11.1. For every x ≥ 19 035 709 163, we have
                                                  
                                            0.15
(3.37)                            x 1−               < θ(x),
                                          (log x)3
and for every x > 1 we have
                                                                     
                                                    0.024334
(3.38)                                 θ(x) < x 1 +                         .
                                                     (log x)3
   Observe that (3.37) is the same as equation (1.4) in [1] and (3.38) improves the constant in (1.5)
of [1] from 0.15 to 0.024334.
Proof. We apply Theorem 1 with k = 3, X0 = 19 035 709 163, and X1 = 1. Since X0 ∈ [1, eJ0 = e27 ],
we proceed as described in Section 3.7.4 and obtain
                m3 = max(A3 (3 400),       max        (Bi∗,3 ), C27,3 , D3 (19 035 709 163, e27 )).
                                        27≤bi ≤3400

Since X1 ≤ eJ = 1019 , we proceed as in Section 3.7.3 and obtain
                                M3 = max(A3 (3 400),            max        (Bbi ,3 )).
                                                             27≤bi ≤3400

Since A3 (3 400) = 2.1719 · 10−2 , max27≤bi ≤3400 (Bbi ,3 ) = BJ,3 = 2.4334 · 10−2 , C27,3 = 5.0536 · 10−2 ,
and D3 (19 035 709 163, e27 ) = 0.15, then
                                   m3 = 0.15 and M3 = 2.4334 · 10−2 .
                                                                                                           
  In our next examples we provide valid proofs of Theorem 4.2 of [15]. The proof of the main
theorem in [15] is incorrect as it applies an incorrect result of Ramaré [39]. The stated leading
constant in Theorem 1.1 of [39] is off by a factor of over 100. The results given here are optimal,
verifiable through computation.
Corollary 11.2. [15, Theorem 4.2] We have
                                                         x
(3.39)                             |ϑ(x) − x| < Mk                for x ≥ X0
                                                       logk x
with
           k     0 1      1           2     2         2           2
           Mk    1 1.2323 0.001       3.965 0.2       0.05        0.01
           X0    1 2      908 994 923 2     3 594 641 112 568 683 7 713 113 853
and
                                                        16
                 k     3     3      3          3           3           4
                 Mk    20.83 10     1          0.78        0.5         151.3
                 X0    2     32 321 89 967 803 158 822 621 767 135 587 2
Proof.

      • Observe that the results with xk ≤ 2 immediately follow from Tables 9, 11, 12, and 13
        below.
      • For m1 , M1 = 0.001, numerical confirmation gives validity for x ∈ [908 994 923, 1010 ).
        Table 13 then gives validity for x ∈ [1010 , 7 · 1011 ), which Table 12 extends to 1019 . From
        here, combining Table 10 and Table 9 gives validity for all x ≥ 1019 .
      • For m2 , M2 = 0.2, numerical confirmation gives validity for x ∈ [3 594 641, 108 ). Table 13
        then gives validity for x ∈ [108 , 7 · 1011 ), which Table 12 extends to 1019 . From here,
        combining Table 10 and Table 9 gives validity for all x ≥ 1019 .
      • For m3 , M3 = 10, numerical confirmation gives validity for x ∈ [32 321, 108 ). Table 13 then
        gives validity for x ∈ [108 , 7 · 1011 ), which Table 12 extends to 1019 . From here, combining
        Table 10 and Table 9 gives validity for all x ≥ 1019 .
      • For m3 , M3 = 1, numerical confirmation gives validity for x ∈ [89 967 803, 108 ). Table 13
        then gives validity for x ∈ [108 , 7 · 1011 ), which Table 12 extends to 1019 . From here,
        combining Table 10 and Table 9 gives validity for all x ≥ 1019 .
      • For m3 , M3 = 0.78, numerical confirmation gives validity for x ∈ [158 882 621, 1010 ). Ta-
        ble 13 then gives validity for x ∈ [1010 , 7 · 1011 ), which Table 12 extends to 1019 . From here,
        combining Table 10 and Table 9 gives validity for all x ≥ 1019 .
      • For m3 , M3 = 0.5, numerical confirmation gives validity for x ∈ [767 135 587, 1010 ). Table 13
        then gives validity for x ∈ [1010 , 7 · 1011 ), which Table 12 extends to 1019 . From here,
        combining Table 10 and Table 9 gives validity for all x ≥ 1019 .
                                                                                                             

Acknowledgements. This research was supported by NSERC Discovery grants of authors H.K.
and N.N.. S.B. was supported by NSERC USRA grants in Summers 2017, 2018, and 2020, K.W.
was supported by an NSERC USRA grant in Summer 2017, and A.L. was supported by a York
University graduate fellowship. Thank-you to Alia Hamieh and Peng-Jie Wong for helping with
the supervision of S.B. and K.W. during part of this project. Thank-you to Andrew Fiori and Josh
Swidinsky for their independent verification of the calculations in Appendix B.

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                                                      19
  Appendix A. Sharper bounds for ψ(x): |ψ(x) − x| < εx where ε is computable for
                                         x ≥ x0 fixed
   In this section we explain how to bound E(x) := (ψ(x) − x)/x. A classic explicit formula that
relates prime numbers to the non-trivial zeros of ζ is given by [8, §17, (1)]:
                                      X xρ                1
(A.1)                      ψ(x) = x −         − log 2π − log(1 − x−2 ),
                                       ρ
                                           ρ              2
when x is not a prime power. The sum over the zeros is not absolutely convergent, hence it is
difficult to directly use this formula to bound E(x). Rosser [40], [41] introduced an averaging
technique to gives bounds for ψ(x). The averaging produces a formula like (A.1), but with an
absolutely convergent sum over zeros of ζ. He later joined forces with Schoenfeld [43] and they
streamlined and improved his arguments. Since their last article in 1975 improvements to their
work have been based on the following:
      1. An improved verification of the partial Riemann hypothesis RH(H): namely all the non-
         trivial zeros ̺ = β + iγ of the Riemann zeta function which have imaginary part |γ| ≤ H
         lie on the 1/2-line.
      2. An improved zero-free region for ζ(s): the zeta function has no zeros in the complex region
                                  1
         {σ + it ∈ C : σ ≥ 1 − R log |t| , |t| ≥ 2}.
      3. A better smooth weight: |ψ(x) − ψϕ (x)| is small, where ψϕ (x) is a smooth variant of ψ(x).
      4. An explicit zero-density result: for σ and T fixed, we define
(A.2)                        N (σ, T ) = #{̺ = β + iγ | β ≥ σ and 0 ≤ γ ≤ T },
         the number of non-trivial zeros ̺ = β + iγ of zeta with β ≥ σ and imaginary part γ between
         0 and T . We have an explicit bound of the form N (σ, T ) ≤ c(σ)T a(σ) (log T )b(σ) , where a, b
         and c are certain functions of σ.
In recent years there has been a lot of activity on bounding E(x) (see [3], [4], [13], [14], [15],
[17], [37]). In 2015 the first theoretical improvements to Rosser and Schoenfeld’s method were
provided by Faber and Kadiri [17]. They introduced the idea of smoothing to the problem and
it was demonstrated that the averaging technique used in [40] and [43] could be interpreted as a
particular case of smoothing. In addition, the use of an explicit zero-density result was first applied
in [17]. Faber and Kadiri’s smooth functions gave better results for x ≥ x0 for any x0 ≤ e4 000 (as
compared with the Rosser and Schoenfeld method used in [14]). In 2016, Büthe [3] used a different
smoothing: he introduced the Logan function which puts more weight on the first zeros for which
RH has been verified. This method did not appeal to a zero-free region or zero-density and it
worked better than [17] for e50 ≤ x0 ≤ e3 000√. In addition, for x ≤ 1019 Büthe [4, Theorem 2] gave
an explicit numerical bound for (ψ(x) − x)/ x. In 2018 Dusart [15] used the Faber-Kadiri method
along with a recent zero-density result of Ramaré [39]. It provided new bounds for ψ(x), θ(x), and
other prime counting sums 3. Recently, Platt and Trudgian employed Perron’s formula along with
the zero-density result of [25] to give the best results in the range x ≥ X0 := e2314 . Currently,
the results of [4], [3], and [37] provide the best explicit bounds for E(x) in the ranges [1, 1019 ],
[1019 , X0 ], and [X0 , ∞) respectively. In this section, we apply the techniques of these articles to
bound E(x).
   The table below displays some of the historical improvements concerning the zeros of zeta that
have been applied in obtaining sharper bounds for ψ(x), θ(x), and π(x).

  3Unfortunately the main theorem of [39] is incorrect and thus bounds claimed in [15] are likely affected, in particular
Theorems 3.2, 3.3 and Table 1 for bounds for ψ(x), and consequently Theorem 4.2 and Table 2 for θ(x). In addition
this unfortunately affects the main theorem [1, Theorem 1.1].
                                                          20
                                                  Table 6

 Author                                H                        R                ε(100)         ε(6 000)
 Rosser & Schoenfeld (1962) [42]       e9.99 ≃ 21 807 [27]      17.52 [42]       9.97 · 10−4    4.92 · 10−6
 Rosser & Schoenfeld (1975) [43]       1 894 438 [43]           9.65 [47] [43]   1.70 · 10−5    3.67 · 10−9
 Dusart (1999) [12]                    545 439 823 [29]         9.65 [47] [43]   9.00 · 10−8    2.41 · 10−9
 Faber & Kadiri (2015) [17] [18]       2 445 999 556 030 [52]   5.70 [23]        2.42 · 10−11   9.68 · 10−14
 Dusart (2016) [14]                    2 445 999 556 030 [52]   5.58 [31]                       6.77 · 10−14
 Büthe (2016) [3]                     -                        -                2.46 · 10−12

Remark. In Table 8 we recorded the best values for ε(b) that we computed using all various methods
known up to today. We found that [3] works best for all b < 2400 at which point [37] works best.
For instance, it gives ε(6 000) = 6.45 · 10−16 , while using Büthe’s theorem gives 1.91 · 10−12 .
——————————————-

A.1. Zeros of the Riemann zeta function. We list here the effective results currently available
for the zeros of the Riemann zeta function and which we use to obtain Theorem 2.
A.1.1. Partial verification of RH. In this article, we use the verification up to
(A.3)                                     H = 2 445 999 556 030
announced by Gourdon [19] and Wedeniwski [52]. In [35] the value of H = 3.061 · 1010 was
established and this was considered the most rigorous verification of RH and a number of articles
preferred to use this value instead of the value in (A.3). After this paper was submitted, an article
of Platt and Trudgian [38] appeared which now verifies that H = 3000175332800 is valid. This
work makes use of the more rigorous techniques as in [35] and thus confirms the earlier calculations
in [19] and [52]. However, the focus of this paper is about the method of computation, and as
Theorem 2 was calibrated so that the verification height for RH is a parameter, we can obtain
results regardless of which height is used. Work in progress, will establish new bounds for ψ(x)
making use of the new values of H from [38]. Moreover, Andrew Fiori and Habiba Kadiri have
developed a webpage which provides bounds for ψ(x) and θ(x) given any set of parameters H, R.
A.1.2. Explicit zero-free region. We use the following type of zero-free region for ζ(s):
Assume there exists R ≥ 1 such that ζ(σ + it) does not vanish when
                                           1
(A.4)                            σ ≥1−            , for every |t| ≥ 2.
                                        R log |t|
We use this with R = 5.573412 as established in [31, Theorem 1].
A.1.3. Explicit zero-density for zeta. We recall that N (σ, T ) is the number of non-trivial zeros in
the region σ ≤ Re(s) ≤ 1 and 0 ≤ Im(s) ≤ T . In [25, Theorem 1.1] the following explicit upper
bounds for N (σ, T ) were established.
                                              9
Theorem 12. [25, Theorem 1.1] Let 10          H ≤ k ≤ 1, d > 0, Y ∈ [1002, H), α > 0, δ ≥ 1, η0 =
0.23622 . . ., 1 + η0 ≤ µ ≤ 1 + η, and η ∈ (η0 , 12 ) be fixed. Let σ > 21 + logdH .
Then there exist C1 , C2 > 0 such that, for any T ≥ H,
                                                C1 (log(kT ))2σ (log T )4(1−σ) T 3 (1−σ) 
                                                                                  8
                      (T − Y )(log T )                                                         C2
         N (σ, T ) ≤                   log 1 +                                              +     (log T )2 ,
(A.5)                       2πd                                  T −Y                         2πd
                       C1                             8          C2
         N (σ, T ) ≤      (log(kT ))2σ (log T )5−4σ T 3 (1−σ) +      (log T )2
                      2πd                                       2πd
                                                     21
where C1 = C1 (α, d, δ, k, Y, σ) and C2 = C2 (d, η, k, Y, µ, σ) are defined in [25, Lemma 4.14].
  From this we have the following corollary.
Corollary 12.1. Let σ ∈ [0.75, 1). Then there exist c1 (σ), c2 (σ) > 0 such that
(A.6)                      N (σ, T ) ≤ c1 (σ)T 8(1−σ)/3 (log T )5−2σ + c2 (σ)(log T )2
where c1 , c2 are given in the table below:

                                                                 8
                Table 7. The bound N (σ, T ) ≤ c1 T 3 (1−σ) (log T )5−2σ + c2 (log T )2 .
                                                                               C1         C2
                            σ0      µ       α         δ          d       c1 = 2πd   c2 = 2πd
                           0.75   1.245   0.189    0.3030      0.338       5.277      4.403
                           0.80   1.245   0.160    0.3030      0.337       6.918      3.997
                           0.85   1.245   0.133    0.3030      0.336       8.975      3.588
                           0.86   1.245   0.127    0.3030      0.335       9.441      3.514
                           0.87   1.245   0.122    0.3030      0.335       9.926      3.430
                           0.88   1.245   0.116    0.3030      0.335      10.431      3.346
                           0.89   1.245   0.111    0.3030      0.335      10.955      3.262
                           0.90   1.245   0.105    0.3030      0.334      11.499      3.186
                           0.91   1.245   0.100    0.3030      0.334      12.063      3.102
                           0.92   1.245   0.095    0.3030      0.334      12.646      3.017
                           0.93   1.245   0.089    0.3030      0.333      13.250      2.941
                           0.94   1.245   0.084    0.3030      0.333      13.872      2.856
                           0.95   1.245   0.079    0.3030      0.333      14.513      2.772
                           0.96   1.245   0.074    0.3030      0.332      15.173      2.694
                           0.97   1.245   0.069    0.3030      0.332      15.850      2.609
                           0.98   1.245   0.064    0.3030      0.331      16.544      2.532
                           0.99   1.245   0.060    0.3030      0.331      17.253      2.446


A.2. Platt and Trudgian’s bounds for ψ(x):

A.2.1. The Prime Number Theorem with a small constant error term. In [37], Platt and Trudgian
use an explicit version of Perron’s formula proven by Dudek [11, Theorem 1.3]: Let x ≥ e1000 and
T satisfies 50 < T ≤ x. Then
                                          X xρ−1                 2
                                                                    
                              ψ(x) − x                 ∗ 2(log x)
(A.7)                                   =           +O
                                 x              ρ             T
                                                  |γ|<T

where A = O∗ (B) means |A| ≤ B. Writing b = log x, we denote
                                                       2b2
(A.8)                                             s0 (b, T ) =
                                                           .
                                                        T
The sum over the zeros is then split vertically at a fixed value 1 − δ with 0.001 ≤ δ ≤ 0.025.
                 X xρ−1                                X xρ−1            X xρ−1
(A.9)                       = Σ1 + Σ2 , with Σ1 =                , Σ2 =           .
                        ρ                                    ρ                 ρ
                   |γ|<T                              |γ|≤T             |γ|≤T
                                                                 β<1−δ                β≥1−δ

The first sum Σ1 is evaluated in [9, Lemma 2.10] by
                                          1                       
(A.10)                       |Σ1 | ≤ x−δ     (log(T /2π))2 + 1.8642 .
                                          2π
                                                          22
We denote
                                                      1                         
(A.11)                        s1 (b, δ, T ) = e−δb         (log(T /2π))2 + 1.8642 .
                                                 2π
To
h   estimate
          i  Σ 2 , an argument of  Pintz [32] is employed.
                                                      j TThe k interval [0, T ] is split into subintervals
                                                         log
   T
      , T where λ > 1, 0 ≤ k ≤ K − 1, and K = log Hλ + 1. Using the zero-free region (A.4) to
  λk+1 λk
bound Re(ρ) we find
                                      X λk+1 x− R log(T /λk )           
                                                       1
                                      K−1
                                                                      T
(A.12)                      |Σ2 | ≤ 2                        N 1 − δ, k .
                                                   T                 λ
                                        k=0
Inserting (A.6) we obtain the following:
                          K−1                                       8δ                                           !
                         λ X k − R log(1T /λk )                T      3
(A.13)         |Σ2 | ≤ 2    λ x                       c1                   (log(T /λk ))3+2δ + c2 (log(T /λk ))2       .
                         T                                     λk
                           k=0
We denote
(A.14) s2 (b, λ, K, T )
       K−1                                        8δ                                      !
     λ X                          b                T 3
  =2        exp k log λ −                      c1        (log(T /λk ))3+2δ + c2 (log(T /λk ))2 .
     T                    R(log T − k log λ)       λk
         k=0
Finally, putting together (A.7), (A.9), (A.10), and (A.13) gives the following result.
Theorem                                                                                 b1
    j T 13.
          k Let b1 , b2 satisfy 1000 ≤ b1 < b2 . Let 0.001 ≤ δ ≤ 0.025, λ > 1, H < T < e , and
     log
K = log Hλ + 1. Then for all x ∈ [eb1 , eb2 ]
                         ψ(x) − x
(A.15)                            ≤ s0 (b2 , T ) + s1 (b1 , δ, T ) + s2 (b1 , δ, λ, K, T ),
                            x
where s0 , s1 , s2 are respectively defined in (A.8), (A.11), and (A.14).
A.2.2. The Prime Number Theorem with an error term of the form (3.3). Let x0 ≥ 1000 be fixed,
and let R be a constant such that Riemann zeta function does not vanish in the region (A.4). We
define
                                                          q
                                         10−16σ log x0 21
(A.16)                 k(σ, x0 ) = (exp(( 3 )( R ) )( logRx0 )5−2σ )−1 ,
                                                                1
(A.17)                   c3 (σ, x0 ) = 2 exp(−2( logRx0 ) 2 )(log x0 )2 k(σ, x0 ),
(A.18)                   c4 (σ, x0 ) = xσ−1
                                        0   ( 2 log x0
                                                 πR + 1.8642)k(σ, x0 ),
                                                                               1
(A.19)                   c5 (σ, x0 ) = 8.01 · c2 (σ) exp(−2( logRx0 ) 2 ) logRx0 k(σ, x0 ).
With the constants defined, we now set
(A.20)            A(σ, x0 ) = 2.0025 · 25−2σ · c1 (σ) + c3 (σ, x0 ) + c4 (σ, x0 ) + c5 (σ, x0 ),

(A.21)                                B = 52 − σ, and C = 16σ  10
                                                           3 − 3 .
Then we have the following result:
Theorem 14. Let x0 ≥ 1000 and let σ ∈ [0.75, 1). For all x ≥ ex0 ,
                                  ψ(x) − x                                 1
(A.22)                                     ≤ A( logR x )B exp(−C( logR x ) 2 )
                                     x
where A, B, and C are defined in (A.20) and (A.21).
                                                           23
  From this and Corollary 5.1 we deduce
Corollary 14.1. Let x0 ≥ 1000. For all x ≥ ex0 ,
                            θ(x) − x                                    1
(A.23)                                ≤ A′ ( logR x )B exp(−C( logR x ) 2 )
                               x
where B and C are defined in (A.21) and
                                        q                                            
(A.24)        A′ = A 1 + A1 ( xR0 )B exp C xR0 a1 (x0 ) exp( −x
                                                              2
                                                                0
                                                                  ) + a  (x
                                                                        2 0 ) exp( −2x0
                                                                                    3   )  ,
where a1 and a2 are defined in Corollary 5.1.
Proof. Let x ≥ ex0 . By writing θ(x) − x = ψ(x) − x + θ(x) − ψ(x), applying the triangle inequality,
and invoking Corollary 5.1, it follows that
(A.25)
    θ(x) − x                            x 21              1                2
             ≤ A( logR x )B exp(−C( log
                                      R  ) ) + a1 (x0 )x− 2 + a2 (x0 )x− 3
       x
                                                                 q                            q         
                                                                      log x                      log x
                                        x 21 
                                                    a  (x
                                                      1 0 ) exp(C       R   )   a  (x
                                                                                  2 0 ) exp(C      R   ) 
             ≤ A( logR x )B exp(−C( log
                                      R ) ) 1 +                              +                          .
                                                          √  log x B                  2
                                                                                               B
                                                        A x R                       Ax 3 logR x
It may be checked the function in brackets decreases for x ≥ ex0 with x0 ≥ 1000 and thus we obtain
(A.23) with A′ given by (A.24).                                                                 
A.3. Bounding ψ(x) using Büthe’s methods. For the range [0, e2313 ], the best bounds for
|ψ(x) − x| are based on two arguments of Büthe [3], [4]. First, for 100 ≤ x ≤ 1019 Büthe [4,
Theorem 2] developed an analytic algorithm to compute ψ(x). Using this, he showed that
                                                  x − ψ(x)
(A.26)                                 − 0.94 ≤      √     ≤ 0.94.
                                                       x
This yields sharp bounds for |ψ(x) − x)| in the range x ≤ 1019 . In [4], Büthe used a smoothing
argument similar to [17]. However, instead he used Logan’s function, which works extremely well
in the range [1019 , e2314 ).
   First, we give a general statement for an application of general bounds of the type (A.26).
Lemma 15. Let B0 , B, and c be positive constants such that
                           x − ψ(x)
(A.27)                        √        ≤c      for all B0 < x ≤ B
                                x
is known. Furthermore, assume for every b0 > 0 there exists ε(b0 ) > 0 such that
(A.28)                            |ψ(x) − x| ≤ ε(b0 )x for all x ≥ eb0 .
Let b be positive such that eb ∈ (B0 , B]. Then, for all x ≥ eb we have
                                                                    
                                  ψ(x) − x              c
(A.29)                                       ≤ max       b
                                                           , ε(log B) .
                                       x               e2
Proof. Multiplying both sides of (A.27) by √1x gives
                              ψ(x) − x      c
                                        ≤ b       for all eb ≤ x ≤ B
                                  x        e2
as √1x ≤ 1b . Then, for x ≥ B we apply (A.28) with b0 = log B. Combining these bounds, we derive
          e2
(A.29).                                                                                                  
                                                    24
  Using [4, (1.5)], we have
Corollary 15.1. Let b be a positive constant such that log 11 < b ≤ 19 log(10). Then we have
                                                          
                     ψ(x) − x            0.94
(A.30)                         ≤ max       b , ε(19 log 10)    for all x ≥ eb .
                         x                e2
Note that by Table 8, we have ε(19 log 10) = 1.93378 · 10−8 .
Proof. By Büthe [4, (1.5)], (A.27) holds with B0 = 11, B = 1019 , and c = 0.94. Thus we may apply
Lemma 15 with B0 = 11, B = 1019 , and c = 0.94 from [4, (1.5)] to obtain (A.30).                
  We now describe the main theorem in [3]. Like [17], this a smoothing argument. Büthe considers
the Fourier transform of Logan’s function which is a sharp cut-off filter kernel described in [28]:
                                                    p
                                              c sin( (ξε)2 − c2 )
                                ℓc,ε (ξ) =         p              .
                                           sinh c    (ξε)2 − c2
Our computations require more values than those provided in [3], so we use his method to compute
more values in these ranges.
Theorem 16. [3, Theorem 1] Let 0 < ε < 10−3 , c ≥ 3, x0 ≥ 100 and α ∈ [0, 1) such that the
inequality
                                             εe−ε x0 |νc (α)|
                                       B0 :=                  >1
                                               2(µc )+ (α)
holds. We denote the zeros of the Riemann zeta function by ρ = β + iγ with β, γ ∈ R. Then, if
β = 21 holds for 0 < γ ≤ εc , the inequality
                                        |ψ(x) − x| ≤ xeεα (E1 + E2 + E3 )
holds for all x ≥ eεα x0 , where
                                                                    
                          2ε       ε 2ε|νc (α)| 2.01ε log log(2x20 )
                   E1 = e log(e x0 )           + √    +                + eεα − 1, 4
                                      log B0       x0      2x0
                            1 + x−1      √
                   E2 = 0.16     0
                                    e0.71 cε log( εc ), and
                             sinh c
                         2    X ℓc,ε(γ)        2
                   E3 = √                  + .
                          x0      c   γ        x 0
                               0<γ< ε


  The νc (α) = νc,1 (α) and µc (α) = µc,1 (α) where νc,ε (α) and µc,ε (α) are defined by [3, p. 2490].




   4This term is written without the eεα − 1 in [3], during personal communication with the author A.L. discovered
this error and are updating the theorem statement to reflect this.
                                                       25
                                     Appendix B. Useful Tables
   The tables in this appendix are subsets of longer tables that can be found in the accompanying
document [2]. Several results in this article make use of the longer tables. All the values are
tabulated using gp-pari and the underlying c libraries. The reported values here are displayed
using a function printf which has automatically chosen to round the values. This rounding is done
differently depending on the compiler used but it effects only the last digit.



                                                                          ′
B.1. Table for ψ(x): |ψ(x) − x| < ε(b, b′ )x for every eb ≤ x ≤ eb . Here b′ is the entry following b
in the table below. The result in the last row is valid in the interval [e25000 , e26000 ]. The calculations
conducted here are using Wedeniwski’s partial verification of the Riemann Hypothesis [52]: H0 =
2 445 999 556 030. Values for b ∈ {20 . . . 2000} are computed using the method of Büthe [3], and
values for b ∈ {2500 . . . 25000} are computed as in Theorem 13 using the method of Platt-Trudgian
[37] .


                                                                                  ′
                        Table 8. |ψ(x) − x| < ε(b, b′ )x for every eb ≤ x ≤ eb .

     b, b′             ε(b, b′ )           b, b′              ε(b, b′ )          b, b′              ε(b, b′ )
  Computed as in [3, Theorem 1]           Computed as in Theorem 13            Computed as in Theorem 13
      20            4.26760 · 10−5        2500            9.06304 · 10−13       14000          1.22655 · 10−35
      21            2.58843 · 10−5        3000            4.59972 · 10−14       15000          4.10696 · 10−37
      22            1.56996 · 10−5        3500            2.48641 · 10−15       16000          1.51402 · 10−38
      23            9.52229 · 10−6        4000            1.42633 · 10−16       17000          6.20397 · 10−40
      24            5.77556 · 10−6        4500            8.68295 · 10−18       18000          2.82833 · 10−41
      25            3.50306 · 10−6        5000            5.63030 · 10−19       19000          1.36785 · 10−42
      30            2.87549 · 10−7        5500            3.91348 · 10−20       20000          7.16209 · 10−44
      35            2.36034 · 10−8        6000            2.94288 · 10−21       21000          4.11842 · 10−45
      40            1.93378 · 10−8        6500            2.38493 · 10−22       22000          2.43916 · 10−46
      45            1.09073 · 10−8        7000            2.07655 · 10−23       23000          1.56474 · 10−47
      50            1.11990 · 10−9        7500            1.96150 · 10−24       24000          1.07022 · 10−48
     100           2.45299 · 10−12        8000            1.97611 · 10−25       25000          7.57240 · 10−50
     200           2.18154 · 10−12        8500            2.12970 · 10−26
     300           2.09022 · 10−12        9000            2.44532 · 10−27
     400           2.03981 · 10−12        9500            2.97001 · 10−28
     500           1.99986 · 10−12        10000           3.78493 · 10−29
     600           1.98894 · 10−12        10500           5.10153 · 10−30
     700           1.97643 · 10−12        11000           7.14264 · 10−31
     800           1.96710 · 10−12        11500           1.04329 · 10−31
     900           1.95987 · 10−12        12000           1.59755 · 10−32
    1000           1.94751 · 10−12        12500           2.53362 · 10−33
    1500           1.93677 · 10−12        13000           4.13554 · 10−34
    2000           1.92279 · 10−12        13500           7.21538 · 10−35



                                                     26
B.2. Sharper bounds for θ(x): x in a large range (x ≥ e2000 ).
                             Ak (b)x
       Table 9. |θ(x) − x| < (log x)k
                                      , for all x ≥ eb , where Ak is defined in Corollary 7.1.

           b        A1 (b)           A2 (b)           A3 (b)           A4 (b)             A5 (b)
         1000   1.1919 · 10−2     1.1919 · 101     1.1919 · 104     1.1919 · 107       1.1919 · 1010
         2000   1.6685 · 10−6    3.3369 · 10−3     6.6738 · 100     1.3348 · 104       2.6696 · 107
         3000   1.3504 · 10−9    4.0512 · 10−6    1.2154 · 10−2     3.6460 · 101       1.0938 · 105
         4000   2.9283 · 10−12   1.1713 · 10−8    4.6852 · 10−5    1.8741 · 10−1       7.4963 · 102
         5000   4.8831 · 10−15   2.4416 · 10−11   1.2208 · 10−7    6.1039 · 10−4       3.0520 · 100
         6000   2.5350 · 10−17   1.5210 · 10−13   9.1257 · 10−10   5.4755 · 10−6      3.2853 · 10−2
         7000   2.1571 · 10−19   1.5100 · 10−15   1.0570 · 10−11   7.3987 · 10−8      5.1791 · 10−4
         8000   2.4895 · 10−21   1.9916 · 10−17   1.5933 · 10−13   1.2747 · 10−9      1.0197 · 10−5
         9000   3.6935 · 10−23   3.3241 · 10−19   2.9917 · 10−15   2.6926 · 10−11     2.4233 · 10−7
        10000   6.7772 · 10−25   6.7772 · 10−21   6.7772 · 10−17   6.7772 · 10−13     6.7772 · 10−9
        11000   1.4933 · 10−26   1.6426 · 10−22   1.8068 · 10−18   1.9875 · 10−14     2.1862 · 10−10
        12000   3.8597 · 10−28   4.6316 · 10−24   5.5579 · 10−20   6.6694 · 10−16     8.0033 · 10−12
        13000   1.1488 · 10−29   1.4934 · 10−25   1.9414 · 10−21   2.5238 · 10−17     3.2809 · 10−13
        14000   3.8771 · 10−31   5.4279 · 10−27   7.5990 · 10−23   1.0639 · 10−18     1.4894 · 10−14
        15000   1.4652 · 10−32   2.1978 · 10−28   3.2967 · 10−24   4.9450 · 10−20     7.4175 · 10−16
        16000   6.1341 · 10−34   9.8146 · 10−30   1.5704 · 10−25   2.5126 · 10−21     4.0201 · 10−17
        17000   2.8193 · 10−35   4.7928 · 10−31   8.1478 · 10−27   1.3852 · 10−22     2.3547 · 10−18
        18000   1.4115 · 10−36   2.5406 · 10−32   4.5731 · 10−28   8.2315 · 10−24     1.4817 · 10−19
        19000   7.6449 · 10−38   1.4526 · 10−33   2.7599 · 10−29   5.2437 · 10−25     9.9630 · 10−21
        20000   4.4536 · 10−39   8.9071 · 10−35   1.7815 · 10−30   3.5629 · 10−26     7.1257 · 10−22
        21000   2.7759 · 10−40   5.8293 · 10−36   1.2242 · 10−31   2.5708 · 10−27     5.3985 · 10−23
        22000   1.8427 · 10−41   4.0538 · 10−37   8.9184 · 10−33   1.9621 · 10−28     4.3165 · 10−24
        23000   1.2974 · 10−42   2.9839 · 10−38   6.8629 · 10−34   1.5785 · 10−29     3.6305 · 10−25
        24000   9.6521 · 10−44   2.3165 · 10−39   5.5596 · 10−35   1.3344 · 10−30     3.2024 · 10−26
        25000   7.5635 · 10−45   1.8909 · 10−40   4.7272 · 10−36   1.1818 · 10−31     3.0000 · 10−27

B.3. Sharper bounds for θ(x): x in a middle range (e20 ≤ x ≤ e25 000 ). We use Corollary 8.1:
                                                               ′
       Table 10. Values for Bk (b, b′ ) in |θ(x) − x| < B(log
                                                          k (b,b )x
                                                              x)k
                                                                    calculated using the method
                                                                              ′
       described in Corollary 8.1. Each Bk (b, b′ ) is valid for eb ≤ x ≤ eb . The last line is
       valid for e24000 ≤ x ≤ e25000 .

          b       B1 (b, b′ )       B2 (b, b′ )      B3 (b, b′ )        B4 (b, b′ )        B5 (b, b′ )
         20     1.8077 · 10−3     3.6154 · 10−2    7.2309 · 10−1       1.4462 · 101        2.9160 · 102
         21     1.1458 · 10−3     2.4062 · 10−2    5.0530 · 10−1       1.0611 · 101        2.2284 · 102
         22     7.2527 · 10−4     1.5956 · 10−2    3.5103 · 10−1       7.7226 · 100        1.6990 · 102
         23     4.5848 · 10−4     1.0545 · 10−2    2.4254 · 10−1       5.5783 · 100        1.2830 · 102
         24     2.8945 · 10−4     6.9468 · 10−3    1.6672 · 10−1       4.0013 · 100        9.6032 · 101
         25     1.8251 · 10−4     4.5626 · 10−3    1.1407 · 10−1       2.8516 · 100        7.1291 · 101
         26     1.1493 · 10−4     2.9882 · 10−3    7.7694 · 10−2       2.0200 · 100        5.2521 · 101
         27     7.2293 · 10−5     1.9519 · 10−3    5.2702 · 10−2       1.4229 · 100        3.8419 · 101
         28     4.5421 · 10−5     1.2718 · 10−3    3.5610 · 10−2      9.9708 · 10−1        2.7918 · 101
         29     2.8507 · 10−5     8.2670 · 10−4    2.3974 · 10−2      6.9525 · 10−1        2.0162 · 101
         30     1.7873 · 10−5     5.3619 · 10−4    1.6086 · 10−2      4.8257 · 10−1        1.4477 · 101
          ..
           .
         43     8.5986 · 10−7     3.7618 · 10−5    1.6458 · 10−3      7.2000 · 10−2        3.1500 · 100
  19 log 10     8.6315 · 10−7     3.7978 · 10−5    1.6711 · 10−3      7.3526 · 10−2
                                                                                           3.2352 · 100
         44     7.8162 · 10−7     3.5173 · 10−5    1.5828 · 10−3      7.1225 · 10−2
                                                                                           3.2052 · 100
                                                                                  Continued on next page

                                                  27
                            Table 10 – continued from previous page
                     ′
    b         B1 (b, b )       B2 (b, b′ )         B3 (b, b′ )        B4 (b, b′ )          B5 (b, b′ )
   45       5.0646 · 10−7    2.3297 · 10   −5
                                                 1.0717 · 10   −3
                                                                    4.9297 · 10−2          2.2677 · 100
   46       3.2935 · 10−7    1.5479 · 10−5       7.2752 · 10−4      3.4194 · 10−2          1.6071 · 100
   47       2.1307 · 10−7    1.0228 · 10   −5
                                                 4.9092 · 10   −4
                                                                    2.3564 · 10−2          1.1311 · 100
    ..
     .
   54       9.8777 · 10−9    5.4328 · 10−7       2.9880 · 10−5      1.6434 · 10−3         9.0388 · 10−2
   55       6.3417 · 10−9    3.5514 · 10−7       1.9888 · 10−5      1.1137 · 10−3         6.2367 · 10−2
   56       4.0668 · 10−9    2.3181 · 10−7       1.3213 · 10−5      7.5315 · 10−4         4.2929 · 10−2
     ..
      .
 2275       4.4153 · 10−9    1.0155 · 10−5       2.3357 · 10−2        5.3721 · 101         1.2356 · 105
 2300       4.4627 · 10−9    1.0376 · 10−5       2.4124 · 10−2        5.6088 · 101         1.3040 · 105
 2325       4.4062 · 10−9    1.0355 · 10−5       2.4333 · 10−2        5.7184 · 101         1.3438 · 105
 2350       4.2245 · 10−9    1.0033 · 10−5       2.3829 · 10−2        5.6593 · 101         1.3441 · 105
 2375       4.0498 · 10−9    9.7196 · 10−6       2.3327 · 10−2        5.5985 · 101         1.3436 · 105
 2400       3.8820 · 10−9    9.4139 · 10−6       2.2829 · 10−2        5.5360 · 101         1.3425 · 105
      ..
       .
 9800      8.4841 · 10−25   8.3992 · 10−21      8.3152 · 10−17     8.2321 · 10−13         8.1497 · 10−9
 9900      5.7395 · 10−25   5.7395 · 10−21      5.7395 · 10−17     5.7395 · 10−13         5.7395 · 10−9
10000      3.8228 · 10−25   3.8610 · 10−21      3.8996 · 10−17     3.9386 · 10−13         3.9780 · 10−9
11100      5.4156 · 10−27   6.0654 · 10−23      6.7933 · 10−19     7.6085 · 10−15        8.5215 · 10−11
12000      1.9330 · 10−28   2.3390 · 10−24      2.8302 · 10−20     3.4245 · 10−16        4.1436 · 10−12
13000      5.5830 · 10−30   7.5370 · 10−26      1.0175 · 10−21     1.3736 · 10−17        1.8544 · 10−13
14000      1.8398 · 10−31   2.7597 · 10−27      4.1396 · 10−23     6.2094 · 10−19        9.3141 · 10−15
15000      6.5711 · 10−33   1.0514 · 10−28      1.6822 · 10−24     2.6915 · 10−20        4.3065 · 10−16
16000      2.5738 · 10−34   4.3755 · 10−30      7.4384 · 10−26     1.2645 · 10−21        2.1497 · 10−17
17000      1.1167 · 10−35   2.0101 · 10−31      3.6182 · 10−27     6.5127 · 10−23        1.1723 · 10−18
18000      5.3738 · 10−37   1.0210 · 10−32      1.9400 · 10−28     3.6859 · 10−24        7.0032 · 10−20
19000      2.7357 · 10−38   5.4714 · 10−34      1.0943 · 10−29     2.1886 · 10−25        4.3771 · 10−21
20000      1.5040 · 10−39   3.1585 · 10−35      6.6328 · 10−31     1.3929 · 10−26        2.9251 · 10−22
21000      9.0605 · 10−41   1.9933 · 10−36      4.3853 · 10−32     9.6476 · 10−28        2.1225 · 10−23
22000      5.6101 · 10−42   1.2903 · 10−37      2.9677 · 10−33     6.8258 · 10−29        1.5699 · 10−24
23000      3.7554 · 10−43   9.0129 · 10−39      2.1631 · 10−34     5.1914 · 10−30        1.2460 · 10−25
24000      2.6755 · 10−44   6.6888 · 10−40      1.6722 · 10−35     4.1805 · 10−31        1.0451 · 10−26


  Table 11. Supplement for Table 10. Values for Bk (b0 ) in |θ(x) − x| < B(logk (b0 )x
                                                                                  x)k
  calculated using the method described in Corollary 8.1 and Bk (b0 ) is defined in
  (3.15). Each Bk (b0 ) is valid for x ∈ [eb0 , e25000 ].

b0            B1 (b0 )         B2 (b0 )           B3 (b0 )            B4 (b0 )              B5 (b0 )
20         1.6844 · 10−3    3.3688 · 10−2      6.7375 · 10−1       5.7184 · 101          1.3441 · 105
21         1.0684 · 10−3    2.2435 · 10−2      4.7114 · 10−1       5.7184 · 101          1.3441 · 105
22         6.7654 · 10−4    1.4884 · 10−2      3.2746 · 10−1       5.7184 · 101          1.3441 · 105
23         4.2780 · 10−4    9.8392 · 10−3      2.2631 · 10−1       5.7184 · 101          1.3441 · 105
24         2.7011 · 10−4    6.4827 · 10−3      1.5559 · 10−1       5.7184 · 101          1.3441 · 105
25         1.7500 · 10−4    4.3750 · 10−3      1.0938 · 10−1       5.7184 · 101          1.3441 · 105
26         1.1022 · 10−4    2.8655 · 10−3      7.4503 · 10−2       5.7184 · 101          1.3441 · 105
27         6.9322 · 10−5    1.8717 · 10−3      5.0536 · 10−2       5.7184 · 101          1.3441 · 105
28         4.3555 · 10−5    1.2196 · 10−3      3.4148 · 10−2       5.7184 · 101          1.3441 · 105
29         2.7336 · 10−5    7.9272 · 10−4      2.4334 · 10−2       5.7184 · 101          1.3441 · 105
30         1.7139 · 10−5    5.1415 · 10−4      2.4334 · 10−2       5.7184 · 101          1.3441 · 105
 ..
  .
                                                                                  Continued on next page
                                               28
                                   Table 11 – continued from previous page
      b0           B1 (b0 )           B2 (b0 )            B3 (b0 )           B4 (b0 )            B5 (b0 )
      43        8.6315 · 10−7      3.7979 · 10−5      2.4334 · 10−2       5.7184 · 101        1.3441 · 105
      43        8.6315 · 10−7      3.7979 · 10−5      2.4334 · 10−2       5.7184 · 101        1.3441 · 105
      44        7.8163 · 10−7      3.5174 · 10 −5
                                                      2.4334 · 10  −2
                                                                          5.7184 · 101        1.3441 · 105
      45        5.0646 · 10−7      2.3298 · 10 −5
                                                      2.4334 · 10  −2
                                                                          5.7184 · 101        1.3441 · 105
      46        3.2935 · 10−7      1.5480 · 10−5      2.4334 · 10−2       5.7184 · 101        1.3441 · 105
      47        2.1308 · 10−7      1.0376 · 10−5      2.4334 · 10−2       5.7184 · 101        1.3441 · 105
          ..
           .
      54        9.8778 · 10−9      1.0376 · 10−5       2.4334 · 10−2       5.7184 · 101       1.3441 · 105
      55        6.3417 · 10−9      1.0376 · 10−5       2.4334 · 10−2       5.7184 · 101       1.3441 · 105
      56        4.4627 · 10−9      1.0376 · 10−5       2.4334 · 10−2       5.7184 · 101       1.3441 · 105
         ..
          .
     2275       4.4627 · 10−9      1.0376 · 10−5       2.4334 · 10−2       5.7184 · 101       1.3441 · 105
     2300       4.4627 · 10−9      1.0376 · 10−5       2.4334 · 10−2       5.7184 · 101       1.3441 · 105
     2325       4.4063 · 10−9      1.0355 · 10−5       2.4334 · 10−2       5.7184 · 101       1.3441 · 105
     2350       4.2245 · 10−9      1.0034 · 10−5       2.3829 · 10−2       5.6594 · 101       1.3441 · 105
     2375       4.0499 · 10−9      9.7196 · 10−6       2.3328 · 10−2       5.5985 · 101       1.3437 · 105
     2400       3.8821 · 10−9      9.4139 · 10−6       2.2829 · 10−2       5.5360 · 101       1.3425 · 105
        ..
         .
     9800       8.4841 · 10−25     8.3993 · 10−21     8.3153 · 10−17      8.2321 · 10−13      8.1498 · 10−9
     9900       5.7396 · 10−25     5.7396 · 10−21     5.7396 · 10−17      5.7396 · 10−13      5.7396 · 10−9
    10000       3.8228 · 10−25     3.8610 · 10−21     3.8997 · 10−17      3.9387 · 10−13      3.9780 · 10−9
       ..
        .
    11000       7.9284 · 10−27     8.8005 · 10−23     9.7685 · 10−19      1.0844 · 10−14     1.2036 · 10−10
    12000       1.9331 · 10−28     2.3390 · 10−24     2.8302 · 10−20      3.4245 · 10−16     4.1437 · 10−12
    13000       5.5830 · 10−30     7.5371 · 10−26     1.0175 · 10−21      1.3737 · 10−17     1.8544 · 10−13
    14000       1.8399 · 10−31     2.7598 · 10−27     4.1396 · 10−23      6.2094 · 10−19     9.3141 · 10−15
    15000       6.5712 · 10−33     1.0514 · 10−28     1.6823 · 10−24      2.6916 · 10−20     4.3065 · 10−16
    16000       2.5739 · 10−34     4.3756 · 10−30     7.4384 · 10−26      1.2646 · 10−21     2.1497 · 10−17
    17000       1.1168 · 10−35     2.0101 · 10−31     3.6182 · 10−27      6.5127 · 10−23     1.1723 · 10−18
    18000       5.3739 · 10−37     1.0211 · 10−32     1.9400 · 10−28      3.6860 · 10−24     7.0033 · 10−20
    19000       2.7357 · 10−38     5.4714 · 10−34     1.0943 · 10−29      2.1886 · 10−25     4.3772 · 10−21
    20000       1.5041 · 10−39     3.1585 · 10−35     6.6329 · 10−31      1.3929 · 10−26     2.9251 · 10−22
    21000       9.0606 · 10−41     1.9934 · 10−36     4.3853 · 10−32      9.6477 · 10−28     2.1225 · 10−23
    22000       5.6101 · 10−42     1.2904 · 10−37     2.9678 · 10−33      6.8258 · 10−29     1.5700 · 10−24
    23000       3.7554 · 10−43     9.0129 · 10−39     2.1631 · 10−34      5.1915 · 10−30     1.2460 · 10−25
    24000       1.3804 · 10−43     3.4508 · 10−39     8.6269 · 10−35      2.1568 · 10−30     5.3919 · 10−26
    25000       1.3804 · 10−43     3.4508 · 10−39     8.6269 · 10−35      2.1568 · 10−30     5.3919 · 10−26

B.4. Lower bound for first values of x (x ≤ 1019 ). 5
B.4.1. Lower bound for first values of x ∈ [eJ0 , 1019 ].

                                      C   x
        Table 12. θ(x) − x > − (logb,kx)k for all x ∈ [eb , 1019 ), where Cb,k is defined in (3.25).

                b            Cb,1            Cb,2             Cb,3             Cb,4               Cb,5
                  Calculated using c = 0.8, C = 0.81, each value valid up to 5 · 1010 ≃ e24.635 .
               20       1.68440 · 10−3 3.36880 · 10−2 6.73750 · 10−1       1.34750 · 101     2.69500 · 102
               21       1.06840 · 10−3 2.24350 · 10−2 4.71140 · 10−1       9.89390 · 100     2.07780 · 102
                                                                                    Continued on next page

  51019 ≃ e43.749
                                                       29
                                         Table 12 – continued from previous page
                   b              Cb,1             Cb,2            Cb,3             Cb,4               Cb,5
                  22         6.76540 · 10−4 1.48840 · 10−2 3.27450 · 10−1       7.20380 · 100     1.58490 · 102
                  23         4.27800 · 10−4 9.83920 · 10−3 2.26310 · 10−1       5.20500 · 100     1.19720 · 102
                                         −4               −3              −1                0
                  24         2.70120 · 10     6.48290 · 10    1.55590 · 10      3.73410 · 10      8.96190 · 101
                                                                                         12    31.097
                     Calculated using c = 0.88, C = 0.86, each value valid up to 32 · 10 ≃ e          .
            log(5 · 1010 )   2.01560 · 10−4 4.96540 · 10−3 1.22330 · 10−1       3.01350 · 100     7.42380 · 101
                 25          1.70330 · 10−4 4.25830 · 10−3 1.06460 · 10−1       2.66140 · 100     6.65350 · 101
                                         −4               −3              −2                0
                 26          1.10220 · 10     2.86560 · 10    7.45050 · 10      1.93720 · 10      5.03650 · 101
                 27          6.93270 · 10−5 1.87190 · 10−3 5.05400 · 10−2       1.36460 · 100     3.68430 · 101
                 28          4.35580 · 10−5 1.21970 · 10−3 3.41500 · 10−2 9.56180 · 10−1          2.67730 · 101
                                         −5               −4              −2               −1
                 29          2.73380 · 10     7.92780 · 10    2.29910 · 10    6.66730 · 10        1.93360 · 101
                                         −5               −4              −2               −1
                 30          1.71400 · 10     5.14180 · 10    1.54260 · 10    4.62760 · 10        1.38830 · 101
                 31          1.07350 · 10−5 3.32790 · 10−4 1.03170 · 10−2 3.19810 · 10−1          9.91400 · 100
                           Calculated using c = C = 0.94, each value valid up to 1019 ≃ e43.749 .
           log(3.2 · 10 ) 1.02600 · 10−5 3.19040 · 10−4 9.92090 · 10−3 3.08510 · 10−1
                       13
                                                                                                  9.59360 · 100
                                         −6               −4              −3               −1
                 32          6.71750 · 10     2.14960 · 10    6.87870 · 10    2.20120 · 10        7.04380 · 100
                 33          4.38000 · 10−6 1.44540 · 10−4 4.76990 · 10−3 1.57410 · 10−1          5.19440 · 100
                 34          2.73610 · 10−6 9.30270 · 10−5 3.16300 · 10−3 1.07540 · 10−1          3.65640 · 100
                                         −6               −5              −3               −2
                 35          1.70780 · 10     5.97730 · 10    2.09210 · 10    7.32220 · 10        2.56280 · 100
                                         −6               −5              −3               −2
                 36          1.06520 · 10     3.83460 · 10    1.38050 · 10    4.96960 · 10        1.78910 · 100
                 37          6.63850 · 10−7 2.45630 · 10−5 9.08810 · 10−4 3.36260 · 10−2          1.24420 · 100
                 38          4.13450 · 10−7 1.57120 · 10−5 5.97020 · 10−4 2.26870 · 10−2 8.62100 · 10−1
                 39          2.57330 · 10−7 1.00360 · 10−5 3.91400 · 10−4 1.52650 · 10−2 5.95320 · 10−1
                 40          1.60060 · 10−7 6.40240 · 10−6 2.56100 · 10−4 1.02440 · 10−2 4.09750 · 10−1
                 41          9.94970 · 10−8 4.07940 · 10−6 1.67260 · 10−4 6.85740 · 10−3 2.81160 · 10−1
                 42          6.18140 · 10−8 2.59620 · 10−6 1.09040 · 10−4 4.57970 · 10−3 1.92350 · 10−1
                 43          3.83820 · 10−8 1.65050 · 10−6 7.09680 · 10−5 3.05170 · 10−3 1.31220 · 10−1

B.4.2. Numerical Verification.

                Table 13. θ(x) − x > − D(log
                                         k (a,b)x
                                             x)k
                                                  for x ∈ [a, b), where Dk (a, b) is defined in (3.29).

       a                  b         D0 (a, b)        D1 (a, b)           D2 (a, b)        D3 (a, b)       D4 (a, b)       D5 (a, b)
       1               1 · 105   1.00000 · 100    1.23228 · 100       3.96481 · 100    2.08282 · 101    1.51224 · 102   1.30475 · 103
     1 · 105           5 · 105   4.73131 · 10−3   5.53160 · 10−2      6.46725 · 10−1   7.56118 · 100    8.93458 · 101   1.07895 · 103
     5 · 105           1 · 106   1.99799 · 10−3   2.67236 · 10−2      3.57434 · 10−1   4.78076 · 100    6.39437 · 101   8.55260 · 102
     1 · 106           5 · 106   1.67162 · 10−3   2.32393 · 10−2      3.23081 · 10−1   4.49158 · 100    6.24434 · 101   8.68209 · 102
     5 · 106           1 · 107   6.56794 · 10−4   1.02357 · 10−2      1.59515 · 10−1   2.48592 · 100    3.87413 · 101   6.03754 · 102
     1 · 107           5 · 107   5.24943 · 10−4   8.47248 · 10−3      1.36744 · 10−1   2.20703 · 100    3.56210 · 101   5.74917 · 102
     5 · 107           1 · 108   2.16306 · 10−4   3.85492 · 10−3      6.87010 · 10−2   1.22436 · 100    2.18201 · 101   3.88870 · 102
     1 · 108           1 · 109   1.48989 · 10−4   2.74568 · 10−3      5.06111 · 10−2   9.4259 · 10−1    1.74450 · 101   3.26946 · 102
     1 · 109        1 · 1010     4.57993 · 10−5   9.59129 · 10−4      2.00861 · 10−2   4.20644 · 10−1   8.80913 · 100   1.84481 · 102
    1 · 1010        2 · 1010     1.63137 · 10−5   3.77870 · 10−4      8.75253 · 10−3   2.02733 · 10−1   4.69587 · 100   1.08770 · 102
 19 035 709 163     2 · 1010     1.13110 · 10−5   2.67726 · 10−4      6.33697 · 10−3   1.49993 · 10−1   3.55028 · 100   8.40336 · 101
    2 · 1010        5 · 1010     1.03687 · 10−5   2.46002 · 10−4      5.83648 · 10−3   1.38472 · 10−1   3.28531 · 100   7.79452 · 101
    5 · 1010        10 · 1010    7.53086 · 10−6   1.85559 · 10−4      4.57216 · 10−3   1.12657 · 10−1   2.77586 · 100   6.83969 · 101
    10 · 1010       20 · 1010    5.26640 · 10−6   1.33915 · 10−4      3.40521 · 10−3   8.65882 · 10−2   2.20178 · 100   5.59872 · 101
                                                                                             Continued on next page

                                                                 30
                                  Table 13 – continued from previous page
      a             b          D0 (a, b)       D1 (a, b)       D2 (a, b)           D3 (a, b)          D4 (a, b)        D5 (a, b)
   20 · 1010    30 · 1010   3.00664 · 10−6   7.86826 · 10−5    2.05910 · 10−3   5.38859 · 10−2      1.41018 · 100   3.69038 · 101
   30 · 1010    40 · 1010   2.41963 · 10−6   6.39936 · 10−5    1.69249 · 10−3   4.47624 · 10−2      1.18386 · 100   3.16250 · 101
   40 · 1010    50 · 1010   2.62662 · 10−6   7.01926 · 10−5    1.87579 · 10−3   5.01279 · 10−2      1.33959 · 100   3.57987 · 101
   50 · 1010    60 · 1010   1.89356 · 10−6   5.10206 · 10−5    1.37472 · 10−3   3.70409 · 10−2   9.98044 · 10−1     2.68917 · 101
   60 · 1010    70 · 1010   1.75478 · 10−6   4.78305 · 10−5    1.30373 · 10−3   3.55359 · 10−2   9.68610 · 10−1     2.64016 · 101




B.5. Final results for Theorem 1.




B.5.1. Values for Theorem 1 for k = 0.




          Table 14. (1 − m0 )x < θ(x) < (1 + M0 )x for all x > X0 = X1 , where m0 and M0
          are defined in Section 3.6.

     log X0 = log X1         M0               m0                log X0 = log X1         M0                 m0
             20         4.2676 · 10−5    9.1639 · 10−5                700          1.9764 · 10−12     1.9765 · 10−12
             25         3.5031 · 10−6    7.4366 · 10−6               1000          1.9475 · 10−12     1.9476 · 10−12
             30         2.8755 · 10−7    6.0751 · 10−7               2000          1.9228 · 10−12     1.9228 · 10−12
             35         2.3603 · 10−8    4.9766 · 10−8               3000          4.5997 · 10−14     4.5998 · 10−14
             40         1.9338 · 10−8    2.1482 · 10−8               4000          1.4263 · 10−16     1.4264 · 10−16
         19 log 10      1.9338 · 10−8    1.9667 · 10−8               5000          5.6303 · 10−19     5.6303 · 10−19
             45         1.0907 · 10−8    1.1084 · 10−8               7000          2.0765 · 10−23     2.0766 · 10−23
             50         1.1199 · 10−9    1.1344 · 10−9               10000         3.7849 · 10−29     3.7850 · 10−29
             60         1.2215 · 10−11   1.2312 · 10−11              11000         7.1426 · 10−31     7.1427 · 10−31
             70         2.7923 · 10−12   2.7930 · 10−12              12000         1.5975 · 10−32     1.5976 · 10−32
             80         2.6108 · 10−12   2.6108 · 10−12              13000         4.1355 · 10−34     4.1356 · 10−34
             90         2.5213 · 10−12   2.5213 · 10−12           13800.7464       2.5423 · 10−35     2.5424 · 10−35
           100          2.4530 · 10−12   2.4530 · 10−12              15000         4.1070 · 10−37     4.1070 · 10−37
           200          2.1815 · 10−12   2.1816 · 10−12              17000         6.2040 · 10−40     6.2040 · 10−40
           300          2.0902 · 10−12   2.0903 · 10−12              20000         7.1621 · 10−44     7.1621 · 10−44
           400          2.0398 · 10−12   2.0399 · 10−12              22000         2.4392 · 10−46     2.4392 · 10−46
           500          1.9999 · 10−12   1.9999 · 10−12              25000         7.5724 · 10−50     7.5724 · 10−50




                                                          31
B.5.2. Values for Theorem 1 for k ∈ {1, 2, 3, 4, 5}.
                                               mk x
        Table 15. For k ∈ {1, 2, 3, 4, 5}, − (log x)k
                                                      < θ(x) − x for x > X0 and θ(x) − x <
          Mk x
        (log x)k
                 for x > X1 ,where mk and Mk are defined in Section 3.

    log X0 = log X1     m1 = M1            m2 = M2         m3 = M3         m4 = M4        m5 = M5
              0         1.2323 · 100       3.9649 · 100    2.0829 · 101   1.5123 · 102   1.3441 · 105
         log(105 )     5.5316 · 10−2
                                          6.4673 · 10−1    7.5612 · 100   8.9346 · 101   1.3441 · 105
       log(5 · 105 )   2.6724 · 10−2
                                          3.5744 · 10−1    4.7808 · 100   6.3944 · 101   1.3441 · 105
         log(106 )     2.3240 · 10−2      3.2309 · 10−1    4.4916 · 100   6.2444 · 101   1.3441 · 105
                                  −2
       log(5 · 106 )   1.0236 · 10        1.5952 · 10−1    2.4860 · 100   5.7184 · 101   1.3441 · 105
         log(107 )     8.4725 · 10−3
                                          1.3675 · 10−1    2.2071 · 100   5.7184 · 101   1.3441 · 105
       log(5 · 107 )   3.8550 · 10−3
                                          6.8701 · 10−2    1.2244 · 100   5.7184 · 101   1.3441 · 105
         log(108 )     2.7457 · 10−3      5.0612 · 10−2   9.4259 · 10−1   5.7184 · 101   1.3441 · 105
         log(109 )     9.5913 · 10−4
                                          2.0087 · 10−2   4.2065 · 10−1   5.7184 · 101   1.3441 · 105
                                  −4
        log(1010 )     3.7787 · 10        8.7526 · 10−3   2.0274 · 10−1   5.7184 · 101   1.3441 · 105
                                  −4
   log(19035709163)    2.6773 · 10        6.3370 · 10−3   1.5000 · 10−1   5.7184 · 101   1.3441 · 105
      log(2 · 1010 )   2.4601 · 10−4      5.8365 · 10−3   1.3848 · 10−1   5.7184 · 101   1.3441 · 105
      log(5 · 1010 )   1.8556 · 10−4
                                          4.5722 · 10−3   1.1266 · 10−1   5.7184 · 101   1.3441 · 105
                                  −4
        log(1011 )     1.3392 · 10        3.4053 · 10−3   8.6589 · 10−2   5.7184 · 101   1.3441 · 105
      log(2 · 1011 )   7.8683 · 10−5
                                          2.0591 · 10−3   5.3886 · 10−2   5.7184 · 101   1.3441 · 105
      log(3 · 1011 )   7.0193 · 10−5      1.8758 · 10−3   5.0536 · 10−2   5.7184 · 101   1.3441 · 105
      log(4 · 1011 )   7.0193 · 10−5
                                          1.8758 · 10−3   5.0536 · 10−2   5.7184 · 101   1.3441 · 105
                                  −5
      log(5 · 1011 )   6.9322 · 10        1.8717 · 10−3   5.0536 · 10−2   5.7184 · 101   1.3441 · 105
      log(6 · 1011 )   6.9322 · 10−5
                                          1.8717 · 10−3   5.0536 · 10−2   5.7184 · 101   1.3441 · 105
             28        4.3555 · 10−5      1.2196 · 10−3   3.4148 · 10−2   5.7184 · 101   1.3441 · 105
                                  −5
             29        2.7336 · 10        7.9272 · 10−4   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −5
             30        1.7139 · 10        5.1415 · 10−4   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
             31        1.0735 · 10−5      3.3277 · 10−4   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
             32        7.0053 · 10−6      2.2417 · 10−4   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −6
             33        4.3798 · 10        1.4454 · 10−4   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −6
             34        2.7360 · 10        9.3023 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
             35        1.7078 · 10−6      5.9771 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
             36        1.0652 · 10−6      3.8345 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −7
             37        8.6315 · 10        3.7979 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −7
             38        8.6315 · 10        3.7979 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
             39        8.6315 · 10−7      3.7979 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −7
             40        8.6315 · 10        3.7979 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −7
             41        8.6315 · 10        3.7979 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −7
             42        8.6315 · 10        3.7979 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
             43        8.6315 · 10−7      3.7979 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −7
         19 log 10     8.6315 · 10        3.7979 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −7
             44        7.8163 · 10        3.5174 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −7
             45        5.0646 · 10        2.3298 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
             46        3.2935 · 10−7      1.5480 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −7
             47        2.1308 · 10        1.0376 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −7
             48        1.3791 · 10        1.0376 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −8
             49        8.9140 · 10        1.0376 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
             50        5.7545 · 10−8      1.0376 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −9
             55        6.3417 · 10        1.0376 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −9
             60        4.4627 · 10        1.0376 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −9
             65        4.4627 · 10        1.0376 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
             70        4.4627 · 10−9      1.0376 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −9
             80        4.4627 · 10        1.0376 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                  −9
             90        4.4627 · 10        1.0376 · 10−5   2.4334 · 10−2   5.7184 · 101   1.3441 · 105
                                Continued on next page

                                                     32
        Table 15 – continued from previous page
   log X0 = log X1         m1 = M1          m2 = M2          m3 = M3          m4 = M4          m5 = M5
          100            4.4627 · 10−9    1.0376 · 10−5    2.4334 · 10−2     5.7184 · 101     1.3441 · 105
          200            4.4627 · 10−9    1.0376 · 10−5    2.4334 · 10−2     5.7184 · 101     1.3441 · 105
                                    −9
          300            4.4627 · 10      1.0376 · 10−5    2.4334 · 10−2     5.7184 · 101     1.3441 · 105
                                    −9
          400            4.4627 · 10      1.0376 · 10−5    2.4334 · 10−2     5.7184 · 101     1.3441 · 105
          500            4.4627 · 10−9    1.0376 · 10−5    2.4334 · 10−2     5.7184 · 101     1.3441 · 105
          600            4.4627 · 10−9    1.0376 · 10−5    2.4334 · 10−2     5.7184 · 101     1.3441 · 105
                                    −9
          700            4.4627 · 10      1.0376 · 10−5    2.4334 · 10−2     5.7184 · 101     1.3441 · 105
                                    −9
          800            4.4627 · 10      1.0376 · 10−5    2.4334 · 10−2     5.7184 · 101     1.3441 · 105
          900            4.4627 · 10−9    1.0376 · 10−5    2.4334 · 10−2     5.7184 · 101     1.3441 · 105
        1000             4.4627 · 10−9    1.0376 · 10−5    2.4334 · 10−2     5.7184 · 101     1.3441 · 105
                                    −9
        1500             4.4627 · 10      1.0376 · 10−5    2.4334 · 10−2     5.7184 · 101     1.3441 · 105
                                    −9
        2000             4.4627 · 10      1.0376 · 10−5    2.4334 · 10−2     5.7184 · 101     1.3441 · 105
        2500             2.2885 · 10−9    5.7783 · 10−6    1.4591 · 10−2     3.6840 · 101     9.3021 · 104
                                    −10
        3000            1.3915 · 10       4.2091 · 10−7    1.2733 · 10−3     3.8516 · 100     1.1651 · 104
                                    −12
        3500            8.7646 · 10       3.0896 · 10−8    1.0891 · 10−4    3.8390 · 10−1     1.3533 · 103
                                    −13
        4000            5.7410 · 10       2.3108 · 10−9    9.3007 · 10−6    3.7436 · 10−2     1.5068 · 102
        5000            2.8715 · 10−15    1.4645 · 10−11   7.4687 · 10−8    3.8091 · 10−4     1.9426 · 100
        6000            1.7952 · 10−17    1.0951 · 10−13   6.6798 · 10−10   4.0747 · 10−6    2.4856 · 10−2
                                    −19
        7000            1.4744 · 10       1.0468 · 10−15   7.4322 · 10−12   5.2769 · 10−8    3.7466 · 10−4
                                    −21
        8000            1.6007 · 10       1.2966 · 10−17   1.0502 · 10−13   8.5065 · 10−10   6.8903 · 10−6
         9000           2.2253 · 10−23    2.0250 · 10−19   1.8428 · 10−15   1.6769 · 10−11   1.5260 · 10−7
                                    −25
        10000           3.8228 · 10       3.8610 · 10−21   3.8997 · 10−17   3.9387 · 10−13   3.9780 · 10−9
                                    −27
        11000           7.9284 · 10       8.8005 · 10−23   9.7685 · 10−19   1.0844 · 10−14   1.2036 · 10−10
                                    −28
        12000           1.9331 · 10       2.3390 · 10−24   2.8302 · 10−20   3.4245 · 10−16   4.1437 · 10−12
        13000           5.5830 · 10−30    7.5371 · 10−26   1.0175 · 10−21   1.3737 · 10−17   1.8544 · 10−13
                                    −31
        14000           1.8399 · 10       2.7598 · 10−27   4.1396 · 10−23   6.2094 · 10−19   9.3141 · 10−15
                                    −33
        15000           6.5712 · 10       1.0514 · 10−28   1.6823 · 10−24   2.6916 · 10−20   4.3065 · 10−16
        16000           2.5739 · 10−34    4.3756 · 10−30   7.4384 · 10−26   1.2646 · 10−21   2.1497 · 10−17
        17000           1.1168 · 10−35    2.0101 · 10−31   3.6182 · 10−27   6.5127 · 10−23   1.1723 · 10−18
                                    −37
        18000           5.3739 · 10       1.0211 · 10−32   1.9400 · 10−28   3.6860 · 10−24   7.0033 · 10−20
                                    −38
        19000           2.7357 · 10       5.4714 · 10−34   1.0943 · 10−29   2.1886 · 10−25   4.3772 · 10−21
        20000           1.5041 · 10−39    3.1585 · 10−35   6.6329 · 10−31   1.3929 · 10−26   2.9251 · 10−22
        21000           9.0606 · 10−41    1.9934 · 10−36   4.3853 · 10−32   9.6477 · 10−28   2.1225 · 10−23
                                    −42
        22000           5.6101 · 10       1.2904 · 10−37   2.9678 · 10−33   6.8258 · 10−29   1.5700 · 10−24
                                    −43
        23000           3.7554 · 10       9.0129 · 10−39   2.1631 · 10−34   5.1915 · 10−30   1.2460 · 10−25
        24000           2.6756 · 10−44    6.6889 · 10−40   1.6723 · 10−35   4.1806 · 10−31   1.0452 · 10−26
                                    −45
        25000           7.5635 · 10       1.8909 · 10−40   4.7272 · 10−36   1.1818 · 10−31   2.9545 · 10−27



  University of Lethbridge, Department of Mathematics and Computer Science, 4401 University
Drive, Lethbridge, AB T1K 3M4, Canada
  Email address: sam.broadbent@uleth.ca
  Email address: habiba.kadiri@uleth.ca
  Email address: lumley@crm.umontreal.ca
  Email address: nathan.ng@uleth.ca
  Email address: kirsten.wilk@uleth.ca




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