                                                        SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)

                                                                       ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY




arXiv:2204.02588v3 [math.NT] 17 May 2023
                                                    Abstract. We improve the unconditional explicit bounds for the error term in the prime counting
                                                    function ψ(x). In particular, we prove that, for all x > 2, we have
                                                                         |ψ(x) − x| < 9.22022 x (log x)3/2 exp(−0.8476836 log x),
                                                                                                                         p

                                                    and that, for all x ≥ exp(3 000),
                                                                                        |ψ(x) − x| < 4.9678 · 10−15 x.
                                                    This compares to results of Platt and Trudgian (2021) who obtained 4.51 · 10−13 x. Our approach
                                                    represents a significant refinement of ideas of Pintz which had been applied by Platt and Trudgian.
                                                    Improvements are obtained by splitting the zeros into additional regions, carefully estimating all of
                                                    the consequent terms, and a significant use of computational methods. Results concerning π(x) will
                                                    appear in the follow up work [11].


                                                                                           1. Introduction
                                           1.1. Context. The main topic of this paper is to look at bounds on the Chebyshev prime counting
                                           function                                       X
                                                                                 ψ(x) =       log(p).
                                                                                                     pr ≤x

                                           We introduce
                                                                                           ψ(x) − x
                                           (1.1)                                          Eψ (x) =  .
                                                                                              x
                                           The prime number theorem for ψ(x), as proven independently by de la Vallée Poussin and Hadamard
                                           in 1896 says
                                           (1.2)                                              lim Eψ (x) = 0.
                                                                                             x→∞

                                           This was a consequence of the Riemann zeta function not vanishing on the vertical line ℜs = 1. We
                                           recall the other prime counting functions and their associated remainder
                                                               X                  θ(x) − x           X                  π(x) − Li(x)
                                           (1.3)       θ(x) =     log p, Eθ (x) =          , π(x) =     1, and Eπ (x) =        x      ,
                                                                                     x                                       log x
                                                                p≤x                                            p≤x
                                                                                              R x dt
                                           where Li(x) is the integral function Li(x) =        2 log t
                                                                                                       . Equivalent versions of the prime number theorem
                                           state that, for θ(x) and π(x) respectively
                                           (1.4)                                  lim Eθ (x) = 0, and lim Eπ (x) = 0.
                                                                                 x→∞                         x→∞


                                              2020 Mathematics Subject Classification. 11A05, 11N25, 11M06, 11N56, 11M26.
                                              Key words and phrases. Prime Number Theorem, Explicit Formulas, ψ(x), Zero Density.
                                              Andrew Fiori graciously acknowledges both the University of Lethbridge for their financial support (start-up grant
                                           and ULRF) as well as the support of NSERC Discovery Grant RGPIN-2020-05316. Habiba Kadiri acknowledges both
                                           the University of Lethbridge for their financial support (ULRF) as well as the support of NSERC Discovery Grant
                                           RGPIN-2020-06731. Joshua Swidinsky acknowledges receiving financial support through an NSERC USRA and from
                                           the University of Lethbridge for this work. Computations performed in this work were conducted using infrastructure
                                           supported by WestGrid (www.westgrid.ca) and Compute Canada (www.computecanada.ca).
                                                                                                       1
2                            ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


    In 1899, de la Vallée Poussin’s zero free region
                                                            1
                         ζ(s) 6= 0 for ℜs > 1 −                      and some R′ > 0,
                                                     R′ log(|t| + 2)
allowed him to prove that the error term Eψ (x) in the estimate (1.2), satisfies
                                                        p      
(1.5)                              Eψ (x) = O exp − c log x ,
and we have the same estimate for Eθ (x) and Eπ (x).
   Explicit bounds for Eψ (x) go back to both separate and joint work of Rosser and Shoenfeld [32, 33,
34, 35]. Recently a number of different theoretical approaches have been applied to prove bounds of
various forms:
                                       √
     (1) Eψ (x) ≤ a(log x)b exp − c log x for all x ≥ x0 , where a, b, c are computable (see [37, 4, 23,
                                              
         30]).
     (2) Eψ (x) ≤ (logηkx)k for all x ≥ x0 , where ηk is computable in terms of x0 and k (see [9, 23, 2]),
     (3) Eψ (x) ≤ ε0 for all x ≥ x0 , where ε0 is computable in terms of x0 (see [10, 37, 9, 23, 4, 2]).
   While all forms can be valid for all sizes of x, (1) is naturally arising in the explicit formula (via
Perron’s formula) relating primes and zeros of the zeta functions. It is also sharper as x gets larger. In
comparison, methods to obtain (2) have been tuned in to give best bound possible for smaller values of
x. For instance, as it is noted in [2, Appendix A3 ], the bound ε0 calculated following Büthe’s methods
[3, 4, 16] is sharper than the one obtained from [30] for all x ≥ x0 when x0 ≤ e2 313 . Note that the
last form (3) the most commonly used in applications and that it is often verified as a consequence of
either (1) or (2). We refer the reader to [2] for a comprehensive and complete description of how to
obtain such bounds for θ(x)1, no matter the size of x and for values of k that are most widely used.
1.2. Main Results. This paper establishes two types of explicit bounds on ψ(x):
Theorem 1.1. For any x0 with log x0 > 1 000, and all 0.9 < σ2 < 1, 2 ≤ c ≤ 30, and N, K ≥ 1 the
formula ε(x0 ) := ε(x0 , σ2 , c, N, K) as defined in (4.1) gives an effectively computable bound
                                        Eψ (x) ≤ ε(x0 )     for all x ≥ x0 .
Moreover, a collection of values, ε(x0 ) computed with well chosen parameters are provided in Table 5.
Theorem 1.2. Let R ≤ 5.573412 be a value for which ζ(s) has no zeros for Res ≥ 1 − R log1Ims for
Ims ≥ 3. For log(x0 ) > 1 000 and with A(x0 ) defined as in (5.2) we have
                                              log x 3/2         r
                                                                   log x 
(1.6)         Eψ (x) ≤ εasm (x0 , x) = A(x0 )             exp − 2           for all x ≥ x0 .
                                                 R                    R
In particular with R = 5.5666305 we have that each pair (x0 , A(x0 )) from Table 6 gives such a bound.
Remark 1.3. The formula for A(x0 ) from (5.2) generally increases as R decreases. Note that since
this paper was submitted, Mossighoff, Trudgian, and Yang [22] have announced a reduced value of
5.558691 for R. The current tables do not reflect this improvement.
The proofs of Theorems 1.1 and 1.2 are given in Sections 4 and 5 respectively.
Corollary 1.4. For all x > 2 we have
                                                                      p
                             Eψ (x) < 9.22022(log x)3/2 exp(−0.8476836 log x).
The proof of Corollary 1.4 is given in Section 5.
   This result provides a significant improvement to the state of the art with regard to a bound valid
for essentially all x.
Remark 1.5 (Comparison with previous similar results).
   1These bounds for θ(x) are the ones needed for applications, but are actually also valid for ψ(x) as the difference
                                                                                                √
between both is taken care of by the rounding up which has a larger effect than the approximate x difference between
ψ(x) and θ(x).
                        SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)                                   3


    (1) As in [30], we use the partial numerical verification of the Riemann Hypothesis H0 = 3 · 1012
        from [29]. As in [5] we use R = 5.5666305 for the zero-free region following [21, Theorem 1
        and Section 6.1]. That is if ̺ = β + iγ is a non trivial zero of the Riemann zeta function, then
                                               1
(1.7)                                     β=       if |γ| ≤ H0
                                               2
        and
                                                 1
(1.8)                                β ≤1−              if |γ| > H0 .
                                             R log |γ|
   (2) It has recently been discovered that the Hiary [14] subconvexity bound for
                                      |ζ(1/2 + it)| ≤ a1 t1/6 log t,
        with a1 = 0.63 relies on an erroneous explicit version of van der Corput second derivative
        test due to Cheng-Graham [7]. See [24, Footnote 3] for details. After accounting for this
        correction, the constant, a1 , changes to 0.77. Very recently a preprint of Hiary, Patel and
        Yang [15] announced that they have recovered and improved this to obtain a1 = 0.618.
           Because the value 0.63 was used in [18, Theorem 1.1], the error would render many results
        concerning the prime number theorem unreliable (for instance [36, 28, 30, 5]).
           In the present work we shall use the the worse sub-convexity bound for the Riemann zeta
        function, namely 0.77, and still recover, and actually improve on these previous results for
        Eψ (x).
           Additionally, in Table 7 we re-calculate the bounds from [18, Theorem 1.1], again using
        0.77 taking this correction into account. Given the existence of the preprint [15] we have also
        included in Table 7 the results one would obtain using 0.618. Finally, we note that if and when
        the 0.618 value is confirmed the results of Corollary 1.4 will improve to
                                                              p
                  Eψ (x) < 8.99284(log x)3/2 exp(−0.8476836 log x) for all x > 2.
       We note that recent work [25] by Patel and Yang offers a significant asymptotic improvement
                                                             27
       to the sub-convexity bound: |ζ(1/2 + it)| ≤ 66.7t 164 for all t ≥ 3. This could further improve
       zero density results using the methods of [18]. Implementing such improvements is a project
       in progress.
   (3) Theorem 1.1 compares directly to [30, Theorem 1, Table 1], [5, Theorem 1.5, Table 1], and [17,
       Theorem 1.1, Table 1]. For instance, for x0 = e3 000 , the admissible value for ε(x0 ) goes from
       4.51 · 10−13 (from [30]), 1.42 · 10−13 (from [5]), and 3.06 · 10−14 (from [17]), to 4.9678 · 10−15 .
   (4) Theorem 1.2 also compares to [30, Theorem 1]. The later states
                                    log x B          r
                                                          log x 
                       Eψ (x) ≤ A             exp − C              for all x ≥ x0 .
                                       R                     R
       where values for (A, B, C) are given in [30, Table 1]. We prove that the expected values
       B = 3/2 and C = 2 are actually admissible values for all x0 (replacing larger bounds such as
       B = 1.51, C = 1.94 when x0 = e10 000 ), In addition we also reduce the values for A, finding for
       instance directly A = 176.3 (and indirectly, see Corollary 1.4, A = 121.2) instead of 535.4 when
       x0 = e10 000 . Very recently [5, Table 1] provided improvements on the constants A, and slight
       improvements on B and C. For example at x0 = e10 000 they obtained A = 268.6, B = 1.508,
       and C = 1.957.
   (5) Finally, Corollary 1.4, which gives the bound
                                                              p
                     Eψ (x) < 9.2203(log x)1.5 exp(−0.84768 log x) for all x > 2,
        can be compared against bounds obtainable from [30] and [5]. The result is both asymptotically
        better, point wise better, and valid on a larger region, than the previous bounds. For example
        from [30] one can conclude the bounds of
                                                     p
                    27.8755(log x)1.52 exp(−0.80057 log x), for all x ≥ exp(3 000).
4                         ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


         Whereas the recent work of [5] would give
                                                    p
                   19.6454(log x)1.514 exp(−0.815895 log x), for all x ≥ exp(3 000).
         More recently, Johnston and Yang [17] have proven
                                                 p
                     8.86(log x)1.514 exp(−0.8288 log x), for all x ≥ exp(3 000),
                                                 p
                     6.72(log x)1.508 exp(−0.8369 log x), for all x ≥ exp(10 000).
         For larger values of x, they use Korobov-Vinogradov’s zero-free region (as made explicit by
         Ford [12]), so their results improve. For instance, they obtain:
                                                   p
                     23.13(log x)1.503 exp(−0.8659 log x), for all x ≥ exp(100 000).
         We also note the work [38] of Yang who provided an explicit Littlewood type zero-free region.
         This would potentially lead to improvements in the intermediate region between where the
         Korobov-Vinogradov’s zero-free region and the de la Vallée Poussin region are the best.
    A discussion of the sources of the various improvements appears in Section 6.
1.3. Acknowledgements. The authors thank Nathan Ng for his important feedback in the prepara-
tion of this manuscript. We also would like to thank him for providing a correction and improvement to
Dudek’s explicit version of Riemann-Von Mangoldt’s formula [8, Thm 1.3.]. In the end, we do not use
this unpublished result, but instead use Cully-Hugill and Johnston’s [5, Thm 1.2.]. We note however
that either result lead to the same results presented in this article. We are grateful to Cully-Hugill and
Johnston for updating us with their progress and sharing a preprint of their work with us. Finally, we
are thanking the referee for their careful reading of our manuscript.

          2. Preliminary results about the zeros of the Riemann-Zeta function
   We first fix some notation. We recall that ζ(s) denotes the Riemann-Zeta function as given by the
series                                      X
                                    ζ(s) =      n−s for ℜs > 1.
                                             n≥1
It can be analytically continued to the whole complex plane: it has a simple pole of residue 1 at s = 1,
and zeros at negative even integers, the remaining non-trivial zeros being located in the vertical strip
0 < ℜs < 1. Throughout this paper, we denote by ρ = β + iγ such a non-trivial zero of ζ(s) with real
part β and imaginary part γ. The Riemann Hypothesis is the conjecture that all such zeros lie on the
vertical line ℜs = 1/2. Since Riemann, calculations for the first zeros have confirmed this hypothesis.
We denote H0 a verification height of the Riemann Hypothesis; that is, if ρ is a zero of ζ(s), and
|γ| < H0 , then β = 1/2. For example, we may take H0 = 3 · 1012 by the recent work of Platt and
Trudgian [29].
   Let T ≥ 1 and σ ∈ (0, 1). We recall that N (T ) counts the number of zeros of ζ(s) in the region
0 < γ < T and 0 < β < 1, and N (σ, T ) the number of zeros of ζ(s) in the region 0 < γ < T and
σ ≤ β < 1.
2.1. Estimates of partial sums over the zeros using the count of the number of zeros N (T ).
Lemma 2.1. Let 1 < U < V and assume that there exist positive constants b1 , b2 , b3 such that, for
every T ≥ 2,
                                   T        T     7
                          N (T ) −      log     +       ≤ R(T ),
                                    2π      2πe 8
where
                          R(T ) := b1 log T + b2 log log T + b3 .
Then,
                                   X 1
                                            ≤ B1 (U, V )
                                          γ
                                         U≤γ<V
                           SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)                              5


with
                            1          b1 log U + b2                 √           2R(U )
(2.1)       B1 (U, V ) =         +                      log(V /U ) log( V U /(2π)) +       .
                            2π       U log U log(U/2π)                                U
   This easily follows from the calculations following [10, Corollary 2.6], which is based on [34, Lemma
7]. See also [1, Lemma 3] for another more refined estimate.

Remark 2.2. By the classical work of Rosser [32, Theorem 19] in Lemma 2.1 we may take b1 = 0.137,
b2 = 0.443, and b3 = 1.588. We note that these constants can be improved, see for instance the work
of Hasanalizade, Shen, and Wong in [13] who obtained:

(2.2)                            b1 = 0.1038, b2 = 0.2573, and b3 = 9.3675.

In any case, in the present application the difference to the final result is negligible.

   Assuming that we have a list of first zeros on the line, namely verifying 0 < γ < T0 for some fixed
value T0 , then we can refine the estimate from Lemma 2.1:

Corollary 2.3. For each pair T0 , S0 as in Table 1 we have that, for all V > T0 ,
                                      X 1
(2.3)                                        < S0 + B1 (T0 , V ),
                                           γ
                                          0<γ<V

where B1 is defined (2.1).

Proof. Each row in Table 1 represents a strict upper bound for the numerical evaluation of
                                                   X 1
(2.4)                              S0 − 10−10 <           < S0 .
                                                        γ
                                                         0<γ<T0

We use the zeros as computed by PlattP[27] and made available through LMFDB [20]. We then apply
Lemma 2.1 to bound the remain sum T0 ≤γ<V γ1 .                                               



        Table 1. Tight upper bounds for values of reciprocal sums of zeros up to height T0
        using Platt’s tabulations, see (2.4).

                                              T0          S0
                                                   100     0.5922435112
                                                 1 000     2.0286569752
                                                10 000     4.3080354951
                                               100 000     7.4318184970
                                             1 000 000    11.3993199147
                                            10 000 000    16.2106480369
                                           100 000 000    21.8657999924
                                         1 000 000 000    28.3647752011
                                        10 000 000 000    35.7075737123
                                        30 610 046 000    39.5797647802


Remark 2.4. Note that in the current application, S0 + B1 (T0 , V ) is a better approximation of the
sum than B1 (0, V ). However, the difference between these estimates does not significantly impact the
result as they become insignificant part of the error once the factor x−1/2 is introduced. However it
could be valuable if one were to consider much smaller values x.
6                              ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


2.2. Estimates of partial sums over zeros close to the edge ℜs = 1 using the count of the
number of zeros N (σ, T ).
Definition (Zero-density bound). We say that N (σ, T ) satisfies (ZDB) if there exists σ0 > 0 such
that
(2.5)                                   N (σ, T ) ≤ Ñ (σ, T ) for all σ > σ0 ,
where Ñ (σ, T ) is of the form
(2.6)                                  Ñ (σ, T ) = c1 T p (log T )q + c2 (log T )2 ,
for some positive functions c1 = c1 (σ), c2 = c2 (σ), p = p(σ), and q = q(σ) with 0 < p(σ) < 1.
   We also require a preliminary result estimating sums over the zeros close to the 1-line, of the form
                                                       X 1
(2.7)                                  s0 (σ, U, V ) =       .
                                                           γ
                                                                U≤γ<V
                                                                σ≤β<1

This result is a particular case of the studies done in [19], where Kadiri, Lumley, and Ng provide esti-
mates for sums of the form U≤γ<V γν(γ)
                             P
                                         m+1 , where ν is a positive, bounded, continuously differentiable
                                     σ≤β<1
function.
   We recall that the incomplete Gamma function is given by
                                         Z ∞
(2.8)                          Γ(s, x) =     ts−1 e−t dt, for ℜs > 0.
                                                    x
Lemma 2.5. Let T0 ≥ 2 and σ > 0. Assume there exists c1 (σ), c2 (σ), p(σ), and q(σ) for which the
zero-density bound (ZDB) is satisfied for N (σ, T ). Assume σ ≥ 5/8 and T0 ≤ U < V . Then,
(2.9)                                         s0 (σ, U, V ) ≤ B0 (σ, U, V ),
where B0 (σ, U, V ) is given by
                                  (log V )q       (log V )2
               B0 (σ, U, V ) =c1      1−p
                                             + c2
                                    V                V
(2.10)                                  c1
                                 +              (Γ(q + 1, (1 − p) log U ) − Γ(q + 1, (1 − p) log V ))
                                   (1 − p)q+1
                                 + c2 (Γ(3, log U ) − Γ(3, log V )) ,
where in this definition, we write c1 , c2 , p, and q respectively in place of c1 (σ), c2 (σ), p(σ), and q(σ).
Proof. We rewrite s0 using the Stieljtes integral definition, then integrate by parts, and use the bound
N ≤ Ñ :
                   Z V                                                 Z V                                 Z V
                         dN (σ, y)       N (σ, V ) N (σ, U )                N (σ, y)          Ñ (σ, V )        Ñ (σ, y)
   s0 (σ, U, V ) =                   =               −               +                dy  ≤              +                dy
                     U       y               V                U         U       y2                V         U       y2
                                                           Z V                      Z V
                      (log V )q         (log V )2               (log y)q                 (log y)2
                 = c1     1−p
                                 +  c 2             +  c 1          2−p
                                                                         dy  +  c 2                dy
                        V                   V                U    y                  U      y2
                      (log V )q         (log V )2
                 = c1     1−p
                                 + c2               + c1 (Jq,2−p (U ) − Jq,2−p (V )) + c2 (J2,2 (U ) − J2,2 (V )) ,
                        V                   V
where we define the integral Ja,b for a > 0, b > 1 by
                                                                     loga y
                                                               Z ∞
(2.11)                                           Ja,b (T ) =                dy.
                                                                T      yb
We conclude by recognizing
              Z ∞                        Z ∞
                                                                             dt             1
  Ja,b (T ) =      (log y)a y −b dy =                    ta (b − 1)−a e−t           =               Γ(a + 1, (b − 1) log T ),
                T                         (log T )(b−1)                    b −  1     (b −  1)a+1
                                         t                        a                     −t
                                                       t                e
doing the variable change y = e (b−1) (so (log y)a = (b−1) a,y
                                                               −b
                                                                  dy = (b−1) dt).                                         
                          SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)                                                              7


Remark 2.6. We recall some well-known facts about the incomplete Gamma function:
                                 
(2.12)     Γ(3, x) = x2 + 2(x + 1) e−x , and Γ(s, x) ∼ xs−1 e−x as x → ∞, for all s > 1.
We recall that 1 − p > 0. Assuming U, V are large enough, an estimate for B0 is
                                    c1  (log U )q    (log V )q 
                                             1−p
                                                   −p             .
                                  1−p U                 V 1−p
   To apply the above we use the functions Ñ obtained in Kadiri-Lumley-Ng’s [18, Theorem 1.1].
                                                                                                                 9
Theorem 2.7. Let H0 denote a verification height for the Riemann Hypothesis. Let 10
                                                                                 H0 ≤ k ≤ 1, d >
0, H ∈ [1002, H0), α > 0, δ ≥ 1, η0 = 0.23622, 1 + η0 ≤ µ ≤ 1 + η, and η ∈ (η0 , 21 ) be fixed. Let
σ > 21 + logdH0 . Then there exist C1 , C2 > 0 such that, for any T ≥ H0 ,
                                                                                    8
                       (T − H)(log T )            C1 (log(kT ))2σ (log T )4(1−σ) T 3 (1−σ)     C2
(2.13)     N (σ, T ) ≤                   log 1 +                                              +     (log T )2 ,
                              2πd                                  T −H                         2πd
                        C1                              8          C2
(2.14)   N (σ, T ) ≤        (log(kT ))2σ (log T )5−4σ T 3 (1−σ) +      (log T )2 .
                       2πd                                        2πd
where C1 = C1 (α, d, δ, k, H, σ) and C2 = C2 (d, η, k, H, µ, σ) are defined in [18, Lemma 4.14] as
                                            8                    4δ(2σ−1) log log H0   2d(2 log log H0 −log log(kH0 ))
                                                δ(2σ−1)M(k,δ)+                       +                                 + 8d
                                                                                                                          3 +2α
(2.15)    C1 (α, d, δ, k, H, σ) = b12 (H)e 3                        log(kH0 )+2δ                    log H0
                                                                                                                                  ×
                                                                        2δ(2σ−1)
                                                     2(1−σ)+ log2dH + log(kH
                                      U(α, k, H0 )                  0        )+2δ
                                                                              0      V(α, k, δ, H0 )2σ−1 ,
                                                            d     C8 (k, µ)
                                                                   
(2.16)    C2 (d, η, k, H, µ, σ) = C7 (η, H) µ − σ +              +           .
                                                          log H0       2
The precise definitions of the individual terms b12 (H), M (k, δ), U(α, k, H0 ), V(α, k, δ, H0 ), C7 (η, H),
and C8 (k, µ) can be be found respectively in [18, (4.18), (4.19), (4.23), (4.56), (4.63), and (4.75)].
Proof. This is [18, Theorem 1.1] except that we may replace C8 (k, µ) by C8 (k, µ)/2 in (2.16) as pointed
out by [5, Section 6] using [6, Theorem 2]. Additionally we note that one should use an admissible value
of a1 in the subconvexity bound for ζ(1/2 + it) (see Remark 1.5(2)) and one should correspondingly
adjust the value of a2 to be a2 = 6e a1 − ζ(1/2).                                                       

   We will denote the bound in equation (2.14) by Ñ (σ, T ).
Remark 2.8. The original table of results published in [18] used H0 = 30 601 646 000 due to Platt
[27]. In Table 7 we provided updated parameters optimized for H0 = 3 · 1012 based on the more recent
work of Platt-Trudgian [29] and the other updates highlighted above.
   An interesting feature of the formulation of the above theorem is that the optimized results apply to
each sigma separately. It is useful in many applications to be able to treat c1 (σ) and c2 (σ) as constant
on an interval. The following Corollary allows us to do this:
Corollary 2.9. For each σ1 , σ2 , c˜1 , and c˜2 given in Table 8, we have that N (σ, T ) ≤ Ñ (σ, T ) for all
σ1 ≤ σ ≤ σ2 , where
(2.17)                            Ñ (σ, T ) = c˜1 T p(σ) (log T )q(σ) + c˜2 (log T )2 ,
with p(σ) = 38 (1 − σ) and q(σ) = 5 − 2σ. Consequently, one can use Table 8 to define c1 (σ) and c2 (σ)
as piece-wise constant functions which satisfy (ZDB).
Proof. Based on the shape of the formulas (2.15) and (2.16), and for any fixed values α, d, δ, η, k, H, µ
satisfying the conditions of the theorem, we have
            C1 (σ) = C1 (α, d, δ, k, H, σ) = d11 dσ12 and C2 (σ) = C2 (d, η, k, H, µ, σ) = d21 − d22 σ,
for values dij = dij (α, d, δ, η, k, H, µ), which depend on the given parameters α, d, δ, η, k, H, µ but are
independent of σ. For instance it is easy to identify d22 = C7 (η, H). As functions of σ, it is clear that
8                              ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


the extreme values of C1 (σ) and C2 (σ) occur at the endpoints of the interval [σ1 , σ2 ]. By explicitly
computing the values of the dij ’s we may identify which of these endpoints gives each extreme value.
Then for the given parameters are given in Table 8, as are the results of taking the maximum value of
both C1 (σ) and C2 (σ) at these endpoints.                                                            


                                         3. Setting up the argument
3.1. Explicit Perron’s Formula. The starting point for our results is an explicit version of Riemann-
Von Mangoldt’s. We cite here two versions, the first due to Dudek, the second to Cully-Hugill and
Johnston.

Theorem 3.1. [8, Theorem 1.3] Let x > e50 be half an odd integer and suppose that 50 < T < x.
Then
                        ψ(x) − x     X xρ−1            
                                                          (log x)2
                                                                   
                                                     ⋆
(3.1)                              ≤              +O 2               ,
                            x                ρ               T
                                                  |γ|<T

where f = g + O∗ (h) means |f − g| ≤ h.

Theorem 3.2. [5, Theorem 1.2] For any α ∈ (0, 1/2] and ω ∈ [0, 1] there exist constants M and xM
such that for max{51, log x} < T < (xα − 2)/5 and some T ⋆ ∈ [T, 2.45T ],
                                        X x̺      x          
(3.2)                   ψ(x) = x −           + O⋆ M (log x)1−ω for all x ≥ xM .
                                         ⋆
                                           ̺       T
                                       |γ|≤T

An admissible value of log(xM ), α, ω, and M is (40, 1/2, 0, 2.091). More are given in [5, Table 5].

Remark 3.3. We note that the above work [5] of Cully-Hugill and Johnston is currently being peer
reviewed, whereas the earlier articles [8] and [31] have errors2. While these errors can be corrected,
doing so may not recover the claimed results. Dudek’s result was used in Platt and Trudgian’s [30] and
represented a non-negligible part of the error term for ψ(x) there. However, in our present application
we shall eventually take T large enough that this term becomes negligible. As the choice of version we
use would ultimately not impact our results, we have decided to use the worse bound published while
including the more stringent hypothesis of [5, Theorem 1.2]. The bound of Proposition 3.4 is Dudek’s
bound as stated in [8, Thm 1.3] and is worse by a factor of (log x) than Cully-Hugill and Johnston’s.
   It is worth noting that all of the asymptotic results and numerical results are still obtainable if
we instead multiply ε1 by a factor of a 100, or even a factor of log x. In most cases this requires no
additional work. For a comparison between these versions of Perron’s formula for small x, see Table 4.
                                                            √
                                                              x
Proposition 3.4. Let x > e50 and 3 log x < T <               3 . Then

                                        ψ(x) − x   X xρ−1
(3.3)                                            ≤        + ε1 (x, T ),
                                           x          ρ
                                                       |γ|<T

with
                                                                  (log x)2
(3.4)                                            ε1 (x, T ) = 2            .
                                                                     T

    2For example [8, Lemma 2.3] is wrong as stated, and used, in particular ζ ′ (s) is unbounded at negative even integers
                                                                               ζ(s)
where the lemma is applied. Additionally, [8, Section 2.1] references in several places Lemma 2.7, when it needs to use
Lemma 2.8 so as to bound I7 . The obvious correction introduces a factor of 20 onto one of the terms. In [5], Cully-Hugill
and Johnston point out errors in Ramaré’s [31].
                          SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)                                      9


3.2. Splitting the sum over the zeros. Following Proposition 3.4, it remains to estimate the sum
over the zeros. Fixing σ1 and σ2 with 1/2 < σ1 < σ2 < 1, we define
                                                X xβ−1
(3.5)                                   Σba = 2           ,
                                                       γ
                                                       0<γ<T
                                                       a≤β<b

and write
                                         X xβ−1
(3.6)                               2           = Σσ0 1 + Σσσ21 + Σ1σ2 .
                                            γ
                                        0<γ<T

We improve upon this result by splitting our sum in a further way.
Remark 3.5.
  (1) In the work of Platt-Trudgian, Pintz’s method is used to break the sum with σ1 = σ2 . Our
      approach of splitting once further will bring significant savings. This is because it allows us
      to reduce the contribution of the large number of zeros on (or relatively near) the half line by
      keeping them less than σ1 while at the same time allowing us to move σ2 towards the 1-line.
      Moving σ2 towards the 1-line is necessary to effectively deal with the interaction of the shape
      of the zero free region with the zero density bounds. Because of how we tightly (relative to
      the quality of the zero density) bound the contribution between σ1 and σ2 further splitting of
      this region is not beneficial.
  (2) We will ultimately take σ1 = 0.9, this choice is arbitrary and any value between 5/8 and 0.95
      would yield similar numerical and asymptotic results. For smaller values of x (for instance
      those less than e1 000 ) taking σ1 smaller would likely be slightly beneficial.
  (3) We will need to optimize the choice of σ2 , we shall see in Section 5 that asymptotically one
      wants to take σ2 slightly larger than 1 − R√2log x to balance the contribution from Σσσ21 and Σ1σ2 .

  In the following sections, we describe methods for bounding each Σσ0 1 , Σσσ21 , Σ1σ2 sum individually.
To do this will require different facts about the zeros of the zeta function.

3.3. Bounds for Σσ0 1 .
Proposition 3.6. Let σ1 ∈ ( 12 , 1) and let (T0 , S0 ) be taken from Table 1. Then,
                                              X       xβ−1
(3.7)                            Σσ0 1 = 2                 ≤ ε2 (x, σ1 , T0 , T ),
                                                       γ
                                             0<γ<T
                                             0≤β<σ1

where
                                                                                
(3.8)          ε2 (x, σ1 , T0 , T ) = 2x−1/2 (S0 + B1 (T0 , T )) + xσ1 −1 − x−1/2 B1 (H0 , T ),

and B1 defined in (2.1).
Proof. Break Σσ0 1 into two parts, first, the area with 1/2 < β < σ1 and second the part with β ≤ 1/2.
In the first we use that xβ−1 ≤ xσ1 −1 , and in the second that xβ−1 ≤ x−1/2 . For the zeros satisfying
β > 1/2, the partial verification of the Riemann Hypothesis implies γ ≥ H0 . Thus
                X xβ−1            X xβ−1            X xβ−1                X 1               X 1
(3.9) Σσ0 1 = 2             =2                +2                ≤ 2x−1/2         +2xσ1 −1             .
                        γ                 γ                 γ                  γ                    γ
              0<γ<T             0<γ<T                  0<γ<T                         0<γ<T          H0 ≤γ<T
               β<σ1             β≤1/2                 1/2<β<σ1                       β≤1/2         1/2<β<σ1

For the first sum, we recognize that
                        X 1        X 1                X      1   X 1                 X       1
                               =       −                       ≤     −                         .
                             γ       γ                       γ     γ                         γ
                      0<γ<T        0<γ<T          H0 ≤γ<T         0<γ<T         H0 ≤γ<T
                      β≤1/2                        β>1/2                        1/2<β<σ
10                          ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


For the second sum, we use the symmetry of zeros accross ℜs = 1/2: for every zero β + iγ that is
being counted on the left sum, we also count the symmetric zero (1 − β) + iγ on the right sum:
                                        X 1        1 X 1
                                                ≤             .
                                              γ    2        γ
                                             H0 ≤γ<T             H0 ≤γ<T
                                            1/2<β<σ1

Thus (3.9) becomes
                                              X 1                                  X         1
(3.10)                    Σσ0 1 ≤ 2x−1/2          + xσ1 −1 − x−1/2                               .
                                                γ                                              γ
                                            0<γ<T                                  H0 ≤γ<T

We conclude by applying Lemma 2.1 and Corollary 2.3 to bound the above sums over the zeros:
                      X 1                              X 1
(3.11)                      ≤ S0 + B1 (T0 , T ), and          ≤ B1 (H0 , T ).
                          γ                                 γ
                        0<γ<T                                       H0 ≤γ<T

                                                                                                                                 

3.4. Bounds for Σσσ21 . In the following we shall use the notation
(3.12)                               Hσ = max (H0 , exp(1/R(1 − σ))) ,
which is defined so that we have
                                                   N (σ, Hσ ) = 0.
Thus we can rewrite (3.5) as
                                                            X       xβ−1
(3.13)                                         Σba = 2                   .
                                                                     γ
                                                          Ha ≤γ<T
                                                           a≤β<b

We are going to split Σσσ21 along the real axis. We introduce the points σ (n) for 0 ≤ n < N :
                                                                          n
(3.14)                                      σ (n) = σ1 + (σ2 − σ1 )         ,
                                                                          N
and the associated heights
(3.15)                             H (n) = max(H0 , exp(1/R(1 − σ (n) ))).
Remark 3.7. Note that for σ < 1 − R log1 H0 , then Hσ = H0 . Here, R log1 H0 = 5.981 · 10−14 , so we will
often have H (n) = H0 for the points σ (n) considered.
Proposition 3.8. Let N ≥ 2 be an integer. If 5/8 ≤ σ1 < σ2 ≤ 1, T ≥ H0 , then
                                               X        xβ−1
(3.16)                          Σσσ21 = 2                    ≤ ε3 (x, σ1 , σ2 , N, T ),
                                                         γ
                                             0<γ<T
                                            σ1 ≤β<σ2

where
                                               σ2 −σ1
(3.17) ε3 (x, σ1 , σ2 , N, T ) = 2x−(1−σ1 )+
                                                                          
                                                 N      B0 σ1 , Hσ1 , T
                                                                     (σ2 −σ1 )    NX
                                                                                    −1
                                                                                                                         n+1
                                            + 2x−(1−σ1 ) 1 − x−         N                B0 (σ (n) , H (n) , T )x(σ2 −σ1 ) N ,
                                                                                  n=1

where B0 is defined in (2.10).
     Note that the assumption σ1 ≥ 5/8 ensures to satisfy conditions of Theorem 2.7.
                                      SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)                                                                                   11


Proof. We recall that
                                                                                       X        xβ−1
                                                                  Σσσ21 = 2                          .
                                                                                                 γ
                                                                                   H0 ≤γ<T
                                                                                   σ1 ≤β<σ2
                                                                                                                         (n+1)
We split along the real axis at the points σ (n) , so as to use xβ−1 < xσ                                                        −1
                                                                                                                                      on each section:
                                                                    N −1                                (n+1)
                                                                    X                  X           xσ           −1
(3.18)                                             Σσσ21 ≤ 2                                                         ,
                                                                      n=0          (n)
                                                                                                         γ
                                                                               H    <γ<T
                                                                             σ(n) ≤β<σ(n+1)

and we recognize the inner sums to be s0 (σ (n) , H (n) , T ), where s0 is defined in (2.7).
                                            N −1                                                                            
                                            X           (n+1)
(3.19)                          Σσσ21 ≤ 2          xσ            −1
                                                                          s0 (σ (n) , H (n) , T ) − s0 (σ (n+1) , H (n) , T ) .
                                            n=0
Instead of directly bounding the above with
                                                            N −1
                                                            X             (n+1)
                                                        2          xσ             −1
                                                                                       B0 (σ (n) , H (n) , T ),
                                                            n=0
we rearrange and telescope the terms:
    N −1
    X           (n+1)
           xσ           −1
                             s0 (σ (n) , H (n) , T ) − s0 (σ (n+1) , H (n) , T )
                                                                                
2
    n=0
                                                                                                     N −1
           (1)                                      (N )                                             X            (n+1)               (n)
= 2xσ            −1
                      s0 (σ (0) , H (0) , T )−2xσ           −1
                                                                 s0 (σ (N ) , H (N ) , T ) + 2               xσ           −1
                                                                                                                               −xσ          −1
                                                                                                                                                  s0 (σ (n) , H (n) , T ).
                                                                                                                                                 
                                                                                                     n=1
Then we apply Lemma 2.5 to bound each s0 by B0 , and obtain the bound:
                                               N −1                      
                (1)                            X         (n+1)        (n)
            2xσ −1 B0 (σ (0) , H (0) , T ) + 2        xσ       −1
                                                                  − xσ −1 B0 (σ (n) , H (n) , T ).
                                                                           n=1
                                                   (n+1)                     (n)
                                                                                                          (σ2 −σ1 )
                                                                                                                               n+1
We conclude by noting that xσ                                −1
                                                                    − xσ           −1
                                                                                         = x−(1−σ1 ) 1 − x− N         x(σ2 −σ1 ) N and that
σ (1) = σ1 + (σ2N
                −σ1 )
                      .                                                                                                                                                
Remark 3.9.
  (1) Taking N = 1 and T → ∞ gives a very simple bound, see Corollary 3.10. Taking N → ∞
      simply gives a tighter numerical bound to a certain Riemann-Stieltjes integral. We illustrate
      the numerical benefit of increasing N in Table 2 as well as the asymptotic behaviour as T → ∞.
  (2) In order to use the above numerically one needs results on zero densities for many values σ.
      These can be obtained on intervals as in Corollary 2.9. In the numerical results we provide
      we use values that are optimized for each σ. Though it is impractical to list the over 100 000
      values we optimized, the total computation takes less than an hour and thus is reasonable if
      one wants optimal values.

Corollary 3.10. If σ1 ≥ 0.9 we have
                                                                 Σσσ21 ≤ 0.00125994 xσ2−1 .
Proof. Taking N = 1 in Prop. 3.8 we obtain the trivial estimate
                                                            Σσσ21 < 2B0 (σ1 , Hσ1 , T )xσ2 −1 .
Noting in (2.10) that limT →∞ Γ(s, T ) = 0, for all s > 1, then
                                           c1
(3.20)        lim B0 (σ1 , Hσ1 , T ) =            Γ(q + 1, (1 − p) log Hσ1 ) + c2 Γ(3, log Hσ1 ),
             T →∞                      (1 − p)q+1
12                             ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


          Table 2. Comparison of bound ε3 (x, σ1 , σ2 , N, T ) as defined in (3.17) for Σσσ21 using
                              p
          the values T = exp(c log x/R), σ1 = 0.9, σ2 = 0.993.

           c  log x N=1                      N=10                   N=100              N=1 000                N=10 000
           1  5 000 4.3062 e-19              4.9030 e-22            2.4879 e-22        2.3529 e-22            2.3442 e-22
           1 50 000 1.2511 e-155             1.1789 e-158           5.8879 e-159       5.4937 e-159           5.4624 e-159
           2  5 000 7.9441 e-19              7.4857 e-22            3.7416 e-22        3.5343 e-22            3.5211 e-22
           2 50 000 1.2511 e-155             1.1789 e-158           5.8879 e-159       5.4937 e-159           5.4624 e-159
          16 5 000 7.9441 e-19               7.4857 e-22            3.7416 e-22        3.5343 e-22            3.5211 e-22
          16 50 000 1.2511 e-155             1.1789 e-158           5.8879 e-159       5.4937 e-159           5.4624 e-159


giving
                                        lim B0 (0.9, 3 · 1012 , T ) ≤ 0.000629970.                                               
                                       T →∞

3.5. Bounds for Σ1σ2 . Thanks to the zero free region (1.8), the remaining zeros ̺ = β + iγ satisfying
                                                                        1
β ≥ σ2 and 0 < γ < T , are actually contained in the region β ≤ 1 − R log  γ and Hσ2 ≤ γ < T , where
                                                      1
Hσ is defined in (3.12). Thus we have the bound for Σσ2 , as defined in (3.5):
                                                                                                1
                                                  X              xβ−1            X        x− R log γ
(3.21)                       Σ1σ2 = 2                                 ≤2                             .
                                                                  γ                          γ
                                            Hσ2 ≤γ<T                         Hσ2 ≤γ<T
                                                     1                         β≥σ2
                                         σ2 ≤β≤1− R log γ


Following the approach taken in [30, Section 3] we apply Riemann-Stieltjes integration to obtain the
following.
                                                                         q
                                                                            log x
                                                                                  
Proposition 3.11. Let 5/8 < σ2 ≤ 1, t0 = t0 (σ2 , x) = max Hσ2 , exp          R      and T > t0 . Let
K ≥ 2 and consider (tk )K
                        k=0 a strictly increasing sequence such that tK = T . Then
                                                                                           1
                                                                                      −
                                                  X        xβ−1                x R log t0
                                 Σ1σ2 = 2                       ≤ 2N (σ2 , T )            ,
                                                            γ                    t0
                                              0<γ<T
                                              σ2 ≤β<1

and
                        K−1                x− R log(t1 k−1 )         1        −      1                 !
                                                                  −
                         X                                       x R log tk  x R log(tK−1 )
            Σ1σ2 ≤ 2           N (σ2 , tk )                    −              +               N (σ2 , T ) .
                                                 tk−1               tk            tK−1
                         k=1

     The next corollary fixes how we choose the sequence (tk ).
                                                                                               q
                                                                                                    log x
                                                                                                            
Corollary 3.12. Let 5/8 < σ2 ≤ 1, t0 = t0 (σ2 , x) = max Hσ2 , exp                                    R       , T > t0 . Let K ≥ 2,
λ = (T /t0 )1/K , and consider (tk )K                                  k
                                    k=0 the sequence given by tk = t0 λ . Then

                                                   X           xβ−1
                                       Σ1σ2 = 2                     ≤ ε4 (x, σ2 , K, T ),
                                                                γ
                                                  0<γ<T
                                                  σ2 ≤β<1

where
                                       K−1         1                                                                  1
                                              − R log                                                            −
                                       X     x        tk
                                                                                                              x R(log t0 )
(3.22)        ε4 (x, σ2 , K, T ) = 2                           Ñ (σ2 , tk+1 ) − Ñ (σ2 , tk ) + 2Ñ (σ2 , t1 )              ,
                                                  tk                                                               t0
                                       k=1

and Ñ (σ, T ) satisfy (ZDB) N (σ, T ) ≤ Ñ (σ, T ).
                            SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)                                               13


Proof of Proposition 3.11. Classical Riemann-Stieltjes integration followed by an integration by part
give:
                              1                           1                                          1
                    Z T − R log
                         x      t                  x− R log T
                                                                             Z T              x− R log t 
             1
(3.23)     Σσ2 ≤ 2                dN (σ2 , t) = 2              N (σ2 , T ) −      N (σ2 , t)d                .
                     Hσ2    t                          T                      Hσ2                 t
Note that
                                             1                 1
                                   d  x− R log t  x− R log t  log x              
                                                     =                         −  1
                                  dt      t                 t2       R(log t)2
                                          q                                                      q
                                               log x                                                 log x
                                                                                                          
becomes negative as soon as t > exp              R    . Thus    taking  t 0 = max  H  σ 2 , exp        R     , we have the
bound
                             1                                       1                             1              1
     Z T              x− R log t 
                                                       Z T  − R log
                                                               x       t                  x− R log t0
                                                                                                            x− R log T 
   −      N (σ2 , t)d               ≤ −N (σ2 , T )         d                = N (σ2 , T )               −               .
      Hσ2                 t                             t0         t                            t0              T
Combining this with (3.23), we obtain
                                                                         1
                                                                    −
                                                 1                 x R log t0
                                                Σσ2 ≤ 2N (σ2 , T )            .
                                                                      t0
We refine this bound by splitting along the imaginary axis at the heights tk , taking t0 as above and
tK = T . We can use the bound N (σ2 , t) ≤ N (σ2 , tk+1 ) on each piece, so that
                                                                           1              1
                                         1       K−1                x− R log       −
                               d  x− R log t  X
               Z T                                                            tk
                                                                                   x R log tk+1 
             −      N (σ2 , t)                 ≤     N (σ2 , tk+1 )              −               .
                Hσ2            dt     t                                 tk            tk+1
                                                          k=0
                                               − R log 1t      −    1    
                                                                  R log T
We note that the (K − 1)th term is N (σ2 , T ) x tK−1         −x T
                                                          K−1
                                                                           , so that when combining the
above with (3.23), and shifting the sum, we obtain
                                  K−1
                                   X                   x− R log(t1 k−1 )    −    1       −      1
                                                                            x R log tk  x R log(tK−1 )             
        ε4 (x, σ2 , K, T ) = 2           Ñ (σ2 , tk )                    −             +               Ñ (σ2 , T ) .
                                                            tk−1               tk           tK−1
                                  k=1

We conclude by ‘telescoping’ the sum over k.                                                                             

Remark 3.13. The parameters T and σ2 play a minor role in the main asymptotic of Σ1σ2 :
    (1) With T fixed, as K increases, the bound ε4 (x, σ2 , K, T ) as defined in (3.22) provides a very
        good numerical approximation of the integral expression
                                                 1
                                        Z T − R log
                                            x       t
                                      2               dÑ (σ2 , t).
                                         t0    t
        The use of ‘telescoping’ is crucial in obtaining a good numerical approximation of the Stieltjes
        integral. Table 3 illustrates the convergence of ε4 (x, σ2 , K, T ) as K increases. The tightness
        of this bound is important in numerical results as well as reducing the constant A in our
        asymptotic results. Previous work often bounds the density of zeros by a constant.
    (2) By fixing λ and considering T = t0 λK as a function of K, Table 3 suggests that the bound
        ε4 (x, σ2 , K, t0 λK ) converges as K → ∞. We note that this would provide a tool to obtain both
        accurate numerical and asymptotic bounds for
                                                 X xβ−1
                                                            .
                                                        γ
                                                         H0 ≤γ
                                                        σ2 ≤β<1

         To establish this, one would need to perform an analysis as in Proposition 3.14. As we will
         not use this we omit the details for now.
14                               ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


          Table 3. Comparison of ε4 (x, σ2 , K, T ) as defined in (3.22) with T = tc0 for various
          values of c and x

                 c    log(x)     K=1              K=10              K=100            K=1 000          K=10 000
                2      5 000     2.2105 e-19      3.4577 e-20       3.0121 e-20      2.9709 e-20      2.9669 e-20
                2     50 000     4.1123 e-73      1.7519 e-74       1.4268 e-74      1.3982 e-74      1.3954 e-74
                4      5 000     5.4637 e-18      4.7133 e-20       3.1058 e-20      2.9801 e-20      2.9678 e-20
           .    4     50 000     1.1429 e-70      3.2640 e-74       1.4928 e-74      1.4045 e-74      1.3960 e-74
                8      5 000     4.1362 e-16      1.1462 e-19       3.3022 e-20      2.9983 e-20      2.9696 e-20
                8     50 000     1.0929 e-66      1.4821 e-73       1.6355 e-74      1.4172 e-74      1.3973 e-74
                16     5 000     2.9350 e-13      5.7273 e-19       3.7327 e-20      3.0353 e-20      2.9732 e-20
                16    50 000     1.2372 e-59      1.9513 e-72       1.9679 e-74      1.4430 e-74      1.3998 e-74

      (3) It is also worth noticing that with T > t0 one easily obtains a lower bound for Σ1σ2 of t22 N (σ, t0 )
                                                                                                                       0
                                                                                                                      3
          which, if we assume our current (ZDB), is bounded below by a constant times (logt2t0 ) which is
                                                                                            0
          asymptotically of the same order of magnitude as the upper bound from ε4 . Hence Corollary
          3.12 describes the true asymptotic of this method with our current tools. Significant improve-
          ments therefore require better zero-free regions, better zero densities, or the use of a more
          sophisticated weight, as in the work of Büthe [3].
     To make use of this bound we must establish that it is decreasing in x.
Proposition 3.14. Fix K ≥ 2 and c > 1, and set t0 , T , and σ2 as functions of x defined by
                                    r log x                              2
(3.24)            t0 = t0 (x) = exp            , T = tc0 , and σ2 = 1 −          .
                                         R                              R log t0
Then, with ε4 (x, σ2 , K, T ) as defined in (3.22), we have that as x → ∞,
                                                                4
                                                         3+
                                            (log t0 ) R log t0                 16w1                   c−1
(3.25)     ε4 (x, σ2 , K, T ) = (1 + o(1))C                    , with C = 2c1 e 3R w13 , and w1 = 1 +     ,
                                                     t20                                               K
where c1 is an admissible value for (ZDB) on some interval [σ1 , 1]. Moreover, both ε4 (x, σ2 , K, T ) and
ε4 (x,σ2 ,K,T )t20
    (log t0 )3     are decreasing in x for x > exp(Re2 ).
                            k
Proof. Denoting wk = 1 + K     (c − 1), with w0 = 1, and recalling tk = t0 λk , we have the identities
                                                      r
                                         2              log x
                      log x = R(log t0 ) , log t0 =            , log tk = (log t0 )wk ,
                                                           R
(3.26)                 −     1
                                            −    1
                                                                                       (w −1)2
                     x R(log t0 )          x R log tk
                                                                             
                                                           −(log t0 ) w1 +wk       −2− kw
                  so               = t−2
                                      0  ,            =  e              k      = t 0
                                                                                          k
                                                                                               .
                          t0                  tk
This allows us to rewrite (3.22) as
                                            K−1       2
                                        2 X − (wkw−1)                                     2
(3.27)           ε4 (x, σ2 , K, T ) =           t 0
                                                    k
                                                         Ñ (σ2 , t k+1 ) − Ñ (σ2 , t k ) + 2 Ñ (σ2 , t1 ).
                                        t20                                                 t0
                                           k=1

We now focus on the expression for Ñ as given by (2.17). Since
                    2                8               16                                4
        σ2 = 1 −          , p(σ2 ) = (1 − σ2 ) =           , q(σ2 ) = 5 − 2σ2 = 3 +          ,
                 R log t0            3           3R log t0                          R log t0
then
                                                   16                      4              16                   4
                                             3R log t0
(3.28) Ñ (σ2 , tk+1 ) − Ñ (σ2 , tk ) = c1 tk+1       (log tk+1 )3+ R log t0 − c1 tk3R log t0 (log tk )3+ R log t0
                                                                                           + c2 (log tk+1 )2 − c2 (log tk )2 .
                              SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)                                                                   15


Using (3.26), we can simplify the above expressions:
                 16                        4                     16                     4
           3R log t0                 3+ R log                                     3+ R log
(3.29) c1 tk+1       (log tk+1 )              t0   − c1 tk3R log t0 (log tk )              t   0

                                                                               4
                                                                                     16wk+1 3+ 4         16wk 3+   4    
                                                           = c1 (log t0 )3+ R log t0 e 3R wk+1R log t0 − e 3R wk R log t0 ,
and
                                                                                    2
                                   c2 (log tk+1 )2 − c2 (log tk )2 = c2 (log t0 )2 wk+1 − wk2 .
                                                                                             
(3.30)
Thus (3.28) can be rewritten as
                                                                                                  4
(3.31)                   Ñ (σ2 , tk+1 ) − Ñ (σ2 , tk ) = C1,k (log t0 )3+ R log t0 + C2,k (log t0 )2 ,
with
                            16wk+1 3+ 4         16wk 3+   4    
                                                                              2
                  C1,k = c1 e 3R wk+1R log t0 − e 3R wk R log t0 , C2,k = c2 wk+1 − wk2 .
                                                                                       
(3.32)
In addition we have
                                                                  4
                                                   16w1     3+ R log                          4
                         Ñ (σ2 , t1 ) = c1 e 3R w1                  t 0
                                                                            (log t0 )3+ R log t0 + c2 w12 (log t0 )2 .
Together with (3.31), we deduce that (3.27) can be simplified to
                                                   K−1                2
                            2(log t0 )3 X − (wkw−1)                          4            C2,k 
(3.33) ε4 (x, σ2 , K, T ) =                 t 0
                                                   k
                                                          C1,k (log  t 0 ) R log t0
                                                                                     +
                                t20                                                      (log t0 )
                                        k=1
                                                  2(log t0 )3                                                     c2 w12
                                                                                                        4
                                                                                                                           
                                                                                    4      16w1    3+ R log t0
                                                +                c1 (log t0 )   R log t0
                                                                                         e  3R  w1             +             .
                                                      t20                                                        (log t0 )
            (w −1)2                                             4
          − kw                      4                      3+
Since t0         k
                     , (log t0 ) R log t0 , log1 t0 , and w1 R log t0 all decrease with t0 > ee , then the expression for
ε4 (x,σ2 ,K,T )t20
    (log t0 )3     as well as just ε4 (x, σ2 , K, T ) both decrease with t0 , and thus with x. We can also deduce
that
                             K−1        (wk −1)2
ε4 (x, σ2 , K, T )t20        X      −     wk
                                                                      C2,k                    16w1 3+ 4                     c2 w12          
               4
         3+ R log
                        =2         t0                  C1,k +                    4
                                                                           1+ R log
                                                                                             +2 c1 e 3R w1 R log t0 +                  4
                                                                                                                                 1+ R log
                                                                                                                                                .
(log t0 )         t0
                             k=1                                (log t0 )           t0
                                                                                                                         (log t0 )        t0


Letting x → ∞, this gives the asymptotic announced in (3.25).                                                                                  
Remark 3.15. From the proposition we see that as x → ∞ the value of ε4 (x, σ2 , K, T ) will tend to a
                        16w1
limit value of C = 2c1 e 3R w13 ≈ 90.825, where the parameters values are chosen as in Section 5. Some
of the asymptotic behaviour can be seen in the values of A(x0 ) in Table 6.

                                                       4. Proof of Theorem 1.1
Proof of Theorem 1.1. The verification [29] and the zero-free region [21] allow to take
                              H0 = 3 · 1012 ,             T0 = 30 610 046 000 and R = 5.5666305.
We now apply Proposition 3.4 together with Proposition 3.6, Proposition 3.8, and Proposition 3.11.
By Proposition 3.14, ε4 is decreasing in x, and this is also easily seen for each of ε1 , ε2 , and ε3 . Taking
                                                       T = t0 (σ2 , x0 )c and σ1 = 0.9,
we have then that for all x ≥ x0 ,
(4.1)
  Eψ (x) ≤ ε(x0 , σ2 , c, N, K) := ε1 (x0 , T ) + ε2 (x0 , 0.9, T0 , T ) + ε3 (x0 , 0.9, σ2 , N, T ) + ε4 (x0 , σ2 , K, T ).
To produce Table 5 we shall fix
(4.2)                               c = 3, K = 100 000, and N = ⌈100 000 (σ2 − σ1 )⌉,
16                             ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


and attempt to optimize the choice of σ2 . We have used an optimized version of the zero density
results Theorem 2.7 for each σ considered. Each row of Table 5 reports a numerical computation of
each εi with the given choices of parameters.                                                  
Remark 4.1. The choice of c = 3 was based on some experimentation, for larger values x (say
x > exp(10 000)) a smaller value might improve the result and for smaller values x (say x < exp(2000))
a larger value might improve the result. In both cases the observed improvement was not in the first
few decimal places.
   The choice N = ⌈100 000(σ2 − σ1 )⌉ is entirely for computational reasons, it ensures the collection
of σ for which we need to optimize the zero density results don’t depend on the actual choice of σ2 .
As illustrated in Tables 2 and 3 reducing N and K does not significantly impact the final result.

                                           5. Proof of Theorem 1.2
Proof of Theorem 1.2. As in Section 4, we immediately have that
                         Eψ (x) ≤ ε1 (x, T ) + ε2 (x, σ1 , T0 , T ) + Σσσ21 + ε4 (x, σ2 , K, T ).
However we shall use simpler bounds for Σσσ21 . We shall take
                           p                                           p
         t0 = t0 (x) = exp( log x/R), c = 4, T = T (x) = t40 = exp(4 log x/R),
(5.1)                                                     2          2
         σ1 = 0.9,       and      σ2 = σ2 (x) = 1 −            =1− √        .
                                                      R log t0      R log x
With these choices we may assume that in the definition of ε4 (x, σ2 , K, T ) all values of σ2 lie in [0.9, 1].
Thus we may use the values from Table 8:
                                           c1 = 17.4194 and c2 = 2.9089.
     We immediately obtain the following asymptotics:
                                                                             2           4
      (1) Using the ε1 of Proposition 3.4, we have 2 (logTx) = 2R2 log(t   t40
                                                                              0)
                                                                                 and for log x > 1 000 this is
          less than
                                                            (log t0 )3
                                            1.92998 · 10−9             .
                                                                 t20
                                                                                 
      (2) We have ε2 (x, 0.9, T0 , T ) = 2x−1/2 (S0 +B1 (T0 , T ))+ xσ1 −1 −x−1/2 B1 (H0 , T ) and for log x >
          1 000 this is less than
                                                       (log t0 )3
                                                 10−10             .
                                                          t20
      (3) By Corollary 3.10 we have Σσ0.9
                                       2
                                           < 0.00125994x−σ2 = 0.00125994
                                                                    t20
                                                                         and for log x > 1 000 this is
          less than
                                                       (log t0 )3
                                         6.8979 · 10−6            .
                                                          t20
                                           ε4 (x0 ,σ2 ,K,T )t20 (log t0 )3
      (4) We have ε4 (x, σ2 , K, T ) <          (log t0 )3         t20
                                                                             and we recall that by Proposition 3.14 the
          quantity
                                                     ε4 (x, σ2 , K, T )t20
                                                          (log t0 )3
         is decreasing in x.
It follows from the above that for x > x0 we have
                                                                      (log t0 )3
                                               Eψ (x) < A(x0 )                   ,
                                                                         t20
where
                         ε4 (x0 , σ2 , K, T (x0 ))(t0 (x0 ))2
(5.2)         A(x0 ) =                                        + 1.92998 · 10−9 + 10−10 + 6.8979 · 10−6 .             
                                   (log(t0 (x0 )))3
                        SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)                                  17


Remark 5.1. We note that the formula A(x0 ) depends on R, however it is decreasing in R, and so
each line of Table 6 gives a value of A(x0 ) which is admissible for R = 5.5666305.
   In the theorem we verify that the formula for A(x0 ) is valid for all R < 5.573412, however we see
that upper bound could be relaxed considerably.
   The following table illustrates the effect of using different values of R (note that only the first two
are known to be valid). Note that although A(x0 ) gets worse as R is decreased, the actual values of ε
decrease.
                           R0       log(x0 )    A(x0 )    εasm (x0 , x0 )   ε(x0 )
                      5.57341200      2 500 245.3670 9.3655 e-13 1.1173 e-13
                      5.57341200      5 000 204.0403 5.2857 e-20 3.0972 e-20
                      5.57341200 50 000 138.1490 6.3117 e-75 4.2087 e-75
                      5.56663050      2 500 245.6773 9.1553 e-13 1.0975 e-13
                      5.56663050      5 000 204.2929 5.1120 e-20 2.9943 e-20
                      5.56663050 50 000 138.3136 5.6411 e-75 3.7604 e-75
                      5.50000000      2 500 248.7907 7.3084 e-13 9.1632 e-14
                      5.50000000      5 000 206.8264 3.6693 e-20 2.1427 e-20
                      5.50000000 50 000 139.9638 1.8502 e-75 1.2242 e-75
Lemma 5.2. For all 0 < log x ≤ 2 100 we have that
                                                                p
                             Eψ (x) ≤ 2(log x)3/2 exp(−0.8476836 log x).
Proof. For 0 < log x ≤ 4 the bound we are claiming is trivial. For 4 < log x ≤ 40 the results of [4] give
a much stronger bound. For 20 < log x ≤ 2 100 the results arising from the work of [3] as calculated in
[2, Table 8] give much strong bounds than those claimed here. Denoting by εBü (x0 ) bounds for Eψ (x)
with x > x0 obtained from [2, Table 8] we have
                                                                                  √
                     εBü (exp(40)) = 1.93378 · 10−8 < 2(1 000)3/2 exp(−0.8476836 1 000),
                                                                                   √
              and εBü (exp(1 000)) = 1.94751 · 10−12 < 2(2 100)3/2 exp(−0.8476836 2 100).
                                        √
Using that 2(log x)3/2 exp(−0.8476836 log x) is decreasing for x > exp(40) completes the result. 

Lemma 5.3. For all 2 100 < log x ≤ 200 000 we have that
                                                                   p
                          Eψ (x) ≤ 9.22022(log x)3/2 exp(−0.8476836 log x).
Proof. As in the previous lemma we use that the function we are comparing to is decreasing and
interpolate a lower bound as a step function using explicit numerical results.
   We shall consider a collection of values bi which divide the interval [2 100, 200 000] and verify that
                                       3/2                  
                                   bi+1               2 p
                         121.096             exp −   √     bi+1 > ε(exp(bi )).
                                     R                 R
In order to do in a way which exposes the limit of this method we instead tabulate values
                                             3/2              
                                             R             2 p
(5.3)                           ε(exp(bi ))         exp √      bi ,
                                             bi             R
and bound the ratio
            3                 23                  3/2              p 
         R 2          2 p        bi         −2 p       bi            2 p
(5.4)           exp √    bi+1          exp √    bi =            exp √ ( bi+1 − bi )
        bi+1           R         R           R        bi+1            R
directly. The subdivision we use, the step size for each subdivision, and the maximum encountered in
each interval are given in the following table. The final column gives a lower bound for an A which
would be admissible on this interval and is the rounded up product of the two maximum values.
18                          ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


             Interval          Step Size      Max Eq. (5.3)     at (log x)   Max Eq. (5.4) Bound on A
       2 100 ≤ log x < 6 000      1/2              120.400      (5 999.5)    1.0050            121.002
       6 000 ≤ log x < 7 250      1/5              120.894      (7 249.8)    1.0011            121.027
       7 250 ≤ log x < 8 250     1/10              121.035      (7 914.3)    1.0005            121.096
       8 250 ≤ log x < 9 000      1/5              120.805      (8 250)      1.0009            120.913
      9 000 ≤ log x < 12 000       1               119.416      (9 000)      1.0050            120.014
     12 000 ≤ log x < 100 000      5               113.332      (12 000)     1.0190            115.486
     100 000 ≤ log x < 200 000    100               84.520      (100 000)    1.1420             96.522

We now simply verify that 121.096/5.56663053/2 < 9.22022 to complete the result.                             
                                                                                                      2
Remark 5.4. The fourth column of Table 6 illustrates a sampling of the values ε(x0 ) (logt0t(x      0)
                                                                                                 0 (x0 ))
                                                                                                          3 that

we used for Lemma 5.3, these represent a theoretical limit to the interpolation. These values increase
initially, especially when bi < 4 595 (so that t0 < 3 · 1012 ). For 4 595 < bi < 7914.3 they slowly increase
to the peak of 121.035 which occurs at bi = 7914.3. Note that at 7914.3 the optimal of σ2 is very
                   1
close to 1 − R log(3·1012 ) which explains why there is a transition near this point. Beyond this point the

values decrease.
   One would certainly expect that by refining the intervals used one can bring the result down slightly.
The computations required to verify all of the values used in the proof of Lemma 5.3 already took over
4000 cpu/hours on various machines in the Westgrid computing cluster.
Proof of Corollary 1.4 . The right hand side of the formula
                            Eψ (x) ≤ 9.22022(log x)3/2 exp(−0.8476836 log x)
                                                                     p

comes from Lemma 5.3.
   For all log x > 200 000 Theorem 1.2 gives the result. For 200 000 ≥ log x ≥ 2 100 Lemma 5.3 gives
the result. For 2 100 ≥ log x ≥ 0 Lemma 5.2 gives the result.                                     
Remark 5.5. The bound
(5.5)                      Eψ (x) ≤ εs (x) := ε1 (x, T ) + 2(S0 + B1 (T0 , T ))x−1/2
is valid for any S0 , T0 from Table 1 and T < 3 · 1012 . In Table 4 we illustrate the relative effectiveness
of this method for this region using the various ε1 for different versions of Perron’s formula admissible
for this region (see Remark 3.3). Note that for T fixed the formula is not decreasing in x, and as such,
provides only a bound at x. The values εBü (x) come from the approach described in [3] as implemented
for [2].

                                      6. Sources of improvements
      (1) About the zero-density:
           (a) For each σ needed, we re-optimize the zero density results of [18, Theorem 1.1] in Theorem
                2.7 and use the new RH verification of Platt-Trudgian [29]. The effect of this improvement
                is small but noticeable. Additionally, we implement the improvement to the C8 term of
                [18, Theorem 1.1] as suggested by [5, Section 6] using [6, Theorem 2]. The effect is to
                divide the term C8 by 2. This does not significantly impact the results of the current
                work. We provide a selection of values in Table 7.
           (b) We extend the zero density result of [18] to one which is valid on intervals (see Corollary
                2.9). This result allows us to set σ2 as a function of x in the asymptotic formula. The
                careful selection of σ2 allows us to get C = 2 and B = 3/2 significantly improving the
                asymptotic behaviour of the bound. We provide a selection of values in Table 8.
      (2) About vertically splitting the zeros:
          Splitting the region vertically once at σ tends to result in a tension between optimizing the
          selection of T and of σ. By further splitting the regions at σ1 and σ2 (as explained in Section
          3.2), we eliminate this tension. This renders negligible the contribution of zeros near the half
                    SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)                                   19


    Table 4. Bounds on Eψ (x) for small x using Equation (5.5) compared to methods of
    Büthe, [3], and Perron’s formula from Dudek and Cully-Hugill and Johnston, see [8]
    and [5]. See also Remark 3.3 (we denote the error terms found with their respective
    methods εDu , εCJ , εBü ). Note, these bounds based on Perron’s formula would hold
    only at x, see Remark 5.5.

                        log x     εDu (x)        εCJ (x)        εBü (x)
                          100   6.66667 e-9    1.70765 e-10   2.00970 e-12
                          200   2.66667 e-8    3.41530 e-10   1.76840 e-12
                          300   6.00000 e-8    5.12295 e-10   1.69300 e-12
                          400   1.06667 e-7    6.83060 e-10   1.65600 e-12
                          500   1.66667 e-7    8.53825 e-10   1.63410 e-12
                          600   2.40000 e-7    1.02459 e-9    1.61950 e-12
                          700   3.26667 e-7    1.19536 e-9    1.60920 e-12
                          800   4.26667 e-7    1.36612 e-9    1.60150 e-12
                          900   5.40000 e-7    1.53689 e-9    1.59550 e-12
                        1 000   6.66667 e-7    1.70765 e-9    1.59070 e-12



    line (see bound ε2 for the sum over the zeros Σ0σ1 in (3.8)), thereby allowing larger values for
    both σ2 and T . This reduces the contribution of the zeros beyond σ2 (see bound ε4 for Σ1σ2 in
    (3.22)) and thus improving significantly both numerical and asymptotic results.
(3) About numerically approximating Riemann-Stieltjes integrals:
     (a) In Proposition 3.8 we improve the use of the zero density result as a function of σ to
          reduce the contribution of the zeros between σ1 and σ2 further. We do this by using a
          better approximation of the Riemann-Stieltjes integral defining the sum over the zeros
          Σσσ21 , rather than simply bounding the integrand by a uniform upper bound. Because in
          this case the integral is impractical to evaluate explicitly, we use a ‘telescoping’ strategy
          which yields a numerically tight upper bound. The numerical nature of this optimization
          means this is important only in the numerical results rather than the asymptotic results.
          The effect of this improvement is small but noticeable in ε3 .
     (b) We improve the use of the zero density results as a function of the height of the zero
          (Lemma 2.5 and Proposition 3.11). By using a better approximation of the Riemann-
          Stieltjes integral, rather than simply bounding the integrand by a uniform upper bound,
          we can significantly reduce the over count. The effect of this improvement is significant
          as it impacts ε4 which is the main contribution to the error terms in Theorems 1.1 and
          1.2 (see Table 5 and (5.2).)
(4) About the zeros close to the half line:
    We use a symmetry argument to reduce the bound on the contribution of the zeros near, but
    not on, the half line (see Proposition 3.6).
(5) About reciprocal sums over zeros on the 1/2-line:
    We improve estimates on reciprocal sums over zeros. Firstly in Lemma 2.1 we give a bound
    on reciprocal sums of zeros in an interval. Additionally we use explicit computations with lists
    of zeros to obtain sharp estimates of the reciprocal sum of zeros up to V = 30 610 046 000, see
    Corollary 2.3 and Table 1:
                                               X 1
                                 39.5797 ≤         ≤ 39.5798.
                                                 γ
                                              0<γ<V

    See [1] for more refined estimates. Note, other improvements, particularly the extra splitting
    of regions, see next point, render this improvement less relevant. These improvements would
    be more important for smaller values of x or if one is assuming RH.
20                            ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


     (6) Tightening the gap between numerical and asymptotic bounds:
         We perform a careful analysis to show that the contribution from the zeros near the zero-free
         region (the ε4 term) is decreasing as a function of x (see Proposition 3.14). This allows us to
         obtain asymptotic results similar to our numerical ones. This gives a significant improvement
         in the asymptotic result of Theorem 1.2.
     (7) About Perron’s formula:
         We note that there is an improved version of Perron’s Formula (see Theorem 3.2). However
         the changes which allow us to increase T render this improvement less relevant. For smaller
         x, simply increasing T further would be more beneficial than using an alternative version
         of Perron’s formula. In addition, other methods, such as those of Büthe [3, 4], are already
         significantly better for log x < 2 100. On the other hand, as x takes larger values, one can
         make improvements beyond q the       5th digit by also using lower values of T (rather than using
                                    log x
                                          
         consistently T = exp 3       R    ).

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[26] J. Pintz, On the remainder term of the prime number formula II. On a theorem of Ingham, Acta Arith. 37 (1980),
     209–220.
[27] D. J. Platt, Isolating some non-trivial zeros of zeta, Math. Comp. 86 (2017), no. 307, 2449–2467.
[28] D. J. Platt and T. S. Trudgian, An improved explicit bound on |ζ(1/2 + it)|, J. Number Theory 147 (2015), 842–851.
                           SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)                                          21


[29] D. J. Platt and T. S. Trudgian. The Riemann hypothesis is true up to 3 · 1012 , Bull. Lond. Math. Soc. 53 (2021),
     no. 3, 792–797.
[30] D. J. Platt and T. S. Trudgian, The error term in the prime number theorem, Math. Comp. 90 (2021), no. 328,
     871–881.
[31] O. Ramaré, Modified truncated Perron formulae, Ann. Math. Blaise Pascal 23 (2016), no. 1, 109–128.
[32] J. B. Rosser, Explicit bounds for some functions of prime numbers, Amer. J.Math. 63 (1941),211–232.
[33] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6
     (1962), 64–94.
[34] J. B. Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x). Math. Comp. 29 (1975),
     243–269.
[35] L. Schoenfeld, Sharper bounds for the Chebyshev functions θ(x) and ψ(x), II. Math. Comp. 30 (1976), no. 134,
     337–360.
[36] T. S. Trudgian, A new upper bound for |ζ(1 + it)|, Bull. Aust. Math. Soc. 89 (2014), no. 2, 259–264.
[37] T. S. Trudgian, Updating the error term in the prime number theorem, Ramanujan J. 39 (2016), no. 2, 225–234.
[38] A. Yang, Explicit bounds on ζ(s) in the critical strip and a zero-free region, arXiv:2301.03165.
22                            ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


                                           7. Auxiliary Tables


           Table 5. Table of values for ε(x0 ) = ε(x0 , σ2 , c, N, K) as defined in (4.1) for Theorem
           1.1. with values c = 3, σ1 = 0.9, N = ⌈100 000(σ2 − σ1 )⌉ and K = 100 000, R =
           5.5666305 and using Dudek’s ε1 . Note: σ2 is only optimized to 5 digits.

       log(x0 )      σ2     ε1             ε2                ε3              ε4               ε(x0 )
         1 000    0.99130   6.8931 e-12    2.2179 e-42       1.1486 e-10     1.2595 e-9       1.3812 e-9
         2 000    0.99221   1.6115 e-18    2.5382 e-85       1.0478 e-13     2.3698 e-12      2.4746 e-12
         2 100    0.99227   4.3625 e-19    1.2306 e-89       5.4150 e-14     1.2705 e-12      1.3246 e-12
         2 200    0.99232   1.2152 e-19    5.9424 e-94       2.7737 e-14     6.8202 e-13      7.0976 e-13
         2 300    0.99236   3.4763 e-20    2.8594 e-98       1.4038 e-14     3.6655 e-13      3.8059 e-13
         2 400    0.99241   1.0198 e-20    1.3716 e-102      7.3304 e-15     1.9693 e-13      2.0426 e-13
         2 500    0.99245   3.0626 e-21    6.5602 e-107      3.7746 e-15     1.0595 e-13      1.0972 e-13
         2 600    0.99249   9.4049 e-22    3.1298 e-111      1.9595 e-15     5.7018 e-14      5.8978 e-14
         2 700    0.99253   2.9495 e-22    1.4897 e-115      1.0255 e-15     3.0704 e-14      3.1729 e-14
         2 800    0.99256   9.4362 e-23    7.0758 e-120      5.2650 e-16     1.6561 e-14      1.7087 e-14
         2 900    0.99260   3.0766 e-23    3.3544 e-124      2.7975 e-16     8.9293 e-15      9.2091 e-15
         3 000    0.99263   1.0213 e-23    1.5874 e-128      1.4554 e-16     4.8223 e-15      4.9678 e-15
         4 000    0.99289   3.8012 e-28    8.3087 e-172      2.5203 e-19     1.0769 e-17      1.1021 e-17
         5 000    0.99311   4.4810 e-32    3.9878 e-215      6.0477 e-22     2.9338 e-20      2.9942 e-20
         6 000    0.99334   1.2102 e-35    1.8179 e-258      2.3940 e-24     1.2737 e-22      1.2976 e-22
         7 000    0.99356   6.1586 e-39    8.0082 e-302      1.4021 e-26     8.3760 e-25      8.5162 e-25
         8 000    0.99379   5.1936 e-42    3.4432 e-345      1.3533 e-28     7.6506 e-27      7.7860 e-27
         9 000    0.99417   6.6323 e-45    1.4537 e-388      2.4527 e-30     8.9809 e-29      9.2262 e-29
        10 000    0.99449   1.2006 e-47    6.0512 e-432      3.7257 e-32     1.3316 e-30      1.3688 e-30
        20 000    0.99619   6.4252 e-70    6.3468 e-866      4.0934 e-47     1.8958 e-45      1.9367 e-45
        30 000    0.99693   4.0605 e-87    4.8888 e-1300     1.2153 e-58     6.5467 e-57      6.6682 e-57
        40 000    0.99736   1.1531 e-101   3.3291 e-1734     2.0196 e-68     1.3291 e-66      1.3493 e-66
        50 000    0.99766   1.6581 e-114   2.1204 e-2168     5.6525 e-77     3.6804 e-75      3.7369 e-75
        60 000    0.99787   3.9127 e-126   1.2951 e-2602     8.2972 e-85     6.5977 e-83      6.6806 e-83
        70 000    0.99804   7.7353 e-137   7.6841 e-3037     6.2358 e-92     4.8619 e-90      4.9243 e-90
        80 000    0.99817   8.2566 e-147   4.4645 e-3471     1.2079 e-98     1.1046 e-96      1.1166 e-96
        90 000    0.99828   3.5041 e-156   2.5526 e-3905     6.4784 e-105    6.2867 e-103     6.3515 e-103
       100 000    0.99838   4.7299 e-165   1.4411 e-4339     9.8527 e-111    7.7127 e-109     7.8112 e-109
       200 000    0.99887   8.7978 e-237   3.2889 e-8682     1.0317 e-158    1.2350 e-156     1.2453 e-156
       300 000    0.99908   6.2208 e-292   5.6126 e-13025    1.0986 e-195    2.1996 e-193     2.2106 e-193
       400 000    0.99921   1.7897 e-338   8.5065 e-17368    1.5373 e-226    2.1209 e-224     2.1363 e-224
       500 000    0.99929   1.6709 e-379   1.2083 e-21710    3.2223 e-254    9.6746 e-252     9.7068 e-252
       600 000    0.99935   1.2951 e-416   1.6472 e-26053    3.4804 e-279    1.7998 e-276     1.8032 e-276
       700 000    0.99940   9.4139 e-451   2.1829 e-30396    8.0982 e-302    3.1872 e-299     3.1953 e-299
       800 000    0.99944   1.5480 e-482   2.8336 e-34739    7.0513 e-323    2.0918 e-320     2.0988 e-320
       900 000    0.99947   2.1427 e-512   3.6206 e-39082    5.1196 e-343    2.6418 e-340     2.6470 e-340
     1 000 000    0.99950   1.2150 e-540   4.5688 e-43425    1.9527 e-361    3.9371 e-359     3.9566 e-359
                  SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)                             23


Table 6. Numerical bounds for Eψ (x) from Theorem 1.2: values for A and εasm are
calculated with values c = 4, σ1 = 0.9, c1 = 17.4194, K = 100 000 and R = 5.5666305.
The values ε(x0 ) are from (4.1) with σ2 optimized with up to 7 digits. The final
column gives an approximate lower bound for A(x0 ), see Lemma 5.3, with t0 as in
(3.24).
                                                                                     2
       log(x0 )      A(x0 )     εasm (x0 , x0 )         ε(x0 )   ε(x0 ) (logt0t(x  0)
                                                                                0 (x0 ))
                                                                                         3

         1 000     338.3058   1.8586 e-6          1.3812 e-9                        0.26
         2 000     263.2129   6.1596 e-11         2.4746 e-12                     10.58
         3 000     233.0775   1.9985 e-14         4.9678 e-15                     57.94
         4 000     215.8229   2.1643 e-17         1.1021 e-17                  109.90
         5 000     204.2929   5.1120 e-20         2.9942 e-20                  119.66
         6 000     195.8842   2.1111 e-22         1.2976 e-22                  120.41
         7 000     189.3959   1.3350 e-24         8.5162 e-25                  120.83
         8 000     184.1882   1.1849 e-26         7.7860 e-27                  121.03
         9 000     179.8849   1.3891 e-28         9.2262 e-29                  119.48
        10 000     176.2484   2.0574 e-30         1.3688 e-30                  117.27
        20 000     156.4775   2.9117 e-45         1.9367 e-45                  104.09
        30 000     147.5424   1.0041 e-56         6.6682 e-57                     97.99
        40 000     142.1006   2.0344 e-66         1.3493 e-66                     94.25
        50 000     138.3136   5.6411 e-75         3.7369 e-75                     91.63
        60 000     135.4686   1.0095 e-82         6.6806 e-83                     89.66
        70 000     133.2221   7.4474 e-90         4.9243 e-90                     88.09
        80 000     131.3849   1.6899 e-96         1.1166 e-96                     86.82
        90 000     129.8428   9.6183 e-103        6.3515 e-103                    85.75
       100 000     128.5221   1.1835 e-108        7.8112 e-109                    84.84
       200 000     121.0360   1.8921 e-156        1.2453 e-156                    79.67
       300 000     117.4647   3.3594 e-193        2.2106 e-193                    77.30
       400 000     115.2251   3.2459 e-224        2.1363 e-224                    75.84
       500 000     113.6357   1.4732 e-251        9.7068 e-252                    74.88
       600 000     112.4241   2.7311 e-276        1.8032 e-276                    74.23
       700 000     111.4565   4.8387 e-299        3.1953 e-299                    73.61
       800 000     110.6577   3.1764 e-320        2.0988 e-320                    73.12
       900 000     109.9819   3.9988 e-340        2.6470 e-340                    72.81
           106     109.3992   5.9745 e-359        3.9566 e-359                    72.45
           107     100.5097   1.6190 e-1153       1.0308 e-1153                   64.00
           108      96.0345   2.6351 e-3669       1.6721 e-3669                   60.94
           109      93.6772   4.0506 e-11628      2.6089 e-11628                  60.34
24                         ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


     Department of Mathematics and Statistics, University of Lethbridge, Canada
     Email address: andrew.fiori@uleth.ca, habiba.kadiri@uleth.ca, jds26@sfu.ca
               SHARPER BOUNDS FOR THE CHEBYSHEV FUNCTION ψ(x)                        25


Table 7. Sample of values of C1 , C2 as defined in Theorem 2.7 for H0 = 3 · 1012 ,
k = 1, µ = 1.2362..., and with optimized parameters d, δ, and α.
This table is computed using the subconvexity bound a1 = 0.77, see Remark 1.5(2).

         σ        α        δ        d        C1        c1     C2     c2
       0.600   0.2745   0.3089   0.3411   6.2364    2.9101 11.3289 5.2863
       0.700   0.2166   0.3088   0.3398   10.3372   4.8414 9.5857 4.4894
       0.800   0.1594   0.3089   0.3382   16.5703   7.7978 7.8423 3.6905
       0.900   0.1041   0.3092   0.3360   25.3876   12.0272 6.0986 2.8891
       0.910   0.0988   0.3093   0.3357   26.4101   12.5219 5.9242 2.8088
       0.920   0.0935   0.3094   0.3354   27.4551   13.0286 5.7497 2.7285
       0.930   0.0883   0.3094   0.3351   28.5212   13.5467 5.5753 2.6481
       0.940   0.0831   0.3095   0.3348   29.6068   14.0756 5.4009 2.5677
       0.950   0.0780   0.3096   0.3344   30.7097   14.6145 5.2264 2.4872
       0.960   0.0730   0.3098   0.3341   31.8278   15.1623 5.0520 2.4067
     . 0.970   0.0681   0.3099   0.3337   32.9584   15.7180 4.8775 2.3261
       0.980   0.0633   0.3101   0.3333   34.0986   16.2804 4.7031 2.2455
       0.990   0.0586   0.3103   0.3329   35.2449   16.8481 4.5286 2.1648
       0.991   0.0582   0.3103   0.3329   35.3598   16.9051 4.5111 2.1567
       0.992   0.0577   0.3103   0.3329   35.4746   16.9621 4.4937 2.1486
       0.993   0.0572   0.3103   0.3328   35.5895   17.0192 4.4762 2.1406
       0.994   0.0568   0.3103   0.3328   35.7044   17.0762 4.4588 2.1325
       0.995   0.0563   0.3104   0.3327   35.8193   17.1334 4.4413 2.1244
       0.996   0.0559   0.3104   0.3327   35.9342   17.1905 4.4239 2.1163
       0.997   0.0554   0.3104   0.3326   36.0492   17.2477 4.4064 2.1083
       0.998   0.0550   0.3104   0.3326   36.1641   17.3049 4.3890 2.1002
       0.999   0.0545   0.3104   0.3326   36.2790   17.3621 4.3715 2.0921


The table below is computed using the subconvexity bound a1 = 0.618, see Remark
1.5(2)
         σ       α       δ       d       C1        c1      C2       c2
       0.600 0.2746 0.3117 0.3447 4.3794 2.0222 11.3310 5.2321
       0.700 0.2167 0.3116 0.3434 7.9114 3.6667 9.5879 4.4437
       0.800 0.1595 0.3117 0.3417 13.8214 6.4369 7.8445 3.6533
       0.900 0.1042 0.3120 0.3394 23.0787 10.8209 6.1007 2.8604
       0.910 0.0988 0.3121 0.3392 24.2157 11.3634 5.9263 2.7810
       0.920 0.0935 0.3122 0.3389 25.3914 11.9255 5.7519 2.7015
       0.930 0.0883 0.3123 0.3386 26.6053 12.5070 5.5774 2.6219
       0.940 0.0831 0.3124 0.3382 27.8565 13.1077 5.4030 2.5423
       0.950 0.0781 0.3125 0.3379 29.1439 13.7272 5.2285 2.4627
       0.960 0.0731 0.3126 0.3375 30.4659 14.3650 5.0541 2.3831
       0.970 0.0682 0.3127 0.3372 31.8207 15.0203 4.8796 2.3033
       0.980 0.0634 0.3129 0.3368 33.2058 15.6923 4.7051 2.2235
       0.990 0.0587 0.3131 0.3364 34.6186 16.3799 4.5307 2.1437
       0.991 0.0582 0.3131 0.3363 34.7613 16.4495 4.5132 2.1357
       0.992 0.0577 0.3131 0.3363 34.9042 16.5192 4.4958 2.1277
       0.993 0.0573 0.3131 0.3362 35.0474 16.5890 4.4783 2.1197
       0.994 0.0568 0.3132 0.3362 35.1908 16.6590 4.4609 2.1117
       0.995 0.0564 0.3132 0.3362 35.3344 16.7292 4.4434 2.1037
       0.996 0.0559 0.3132 0.3361 35.4782 16.7994 4.4260 2.0958
       0.997 0.0555 0.3132 0.3361 35.6223 16.8698 4.4085 2.0878
       0.998 0.0550 0.3132 0.3360 35.7666 16.9404 4.3911 2.0798
       0.999 0.0546 0.3133 0.3360 35.9111 17.0110 4.3736 2.0718
26                     ANDREW FIORI, HABIBA KADIRI, JOSHUA SWIDINSKY


     Table 8. Sample of values for c˜1 , c˜2 as in Corollary 2.9 for H0 = 3 · 1012 and k = 1
     with µ = 1.2362 . . . and C7 (η, H0 ) = 17.424 and optimized parameters α, δ, and d.
     This table is computed using the subconvexity bound a1 = 0.77, see Remark 1.5(2).

          σ1     σ2       α         δ        d       C1         c˜1    C2     c˜2
         0.60   0.65   0.2456    0.3089   0.3405   8.0587    3.7669 11.3285 5.2954
         0.65   0.70   0.2167    0.3089   0.3399   10.3373   4.8415 10.4569 4.8975
         0.70   0.75   0.1879    0.3089   0.3391   13.1505   6.1727 9.5853 4.4992
         0.75   0.80   0.1595    0.3089   0.3383   16.5704   7.7979 8.7136 4.1006
         0.80   0.81   0.1538    0.3089   0.3381   17.3322   8.1610 7.8423 3.6926
         0.81   0.82   0.1482    0.3090   0.3379   18.1208   8.5373 7.6679 3.6126
         0.82   0.83   0.1426    0.3090   0.3377   18.9362   8.9269 7.4935 3.5326
         0.83   0.84   0.1370    0.3090   0.3374   19.7785   9.3298 7.3192 3.4526
         0.84   0.85   0.1314    0.3090   0.3372   20.6478   9.7461 7.1448 3.3725
         0.85   0.86   0.1259    0.3091   0.3370   21.5438   10.1759 6.9704 3.2924
         0.86   0.87   0.1204    0.3091   0.3368   22.4663   10.6191 6.7960 3.2123
         0.87   0.88   0.1150    0.3092   0.3365   23.4149   11.0755 6.6216 3.1321
         0.88   0.89   0.1095    0.3092   0.3363   24.3889   11.5450 6.4473 3.0519
         0.89   0.90   0.1041    0.3093   0.3360   25.3877   12.0272 6.2729 2.9717
         0.90   0.91   0.09880   0.3093   0.3357   26.4101   12.5220 6.0984 2.8915
         0.91   0.92   0.09350   0.3094   0.3354   27.4552   13.0287 5.9240 2.8112
         0.92   0.93   0.08830   0.3095   0.3351   28.5213   13.5468 5.7496 2.7309
         0.93   0.94   0.08310   0.3096   0.3348   29.6068   14.0757 5.5752 2.6506
         0.94   0.95   0.07810   0.3097   0.3345   30.7098   14.6145 5.4007 2.5702
         0.95   0.96   0.07310   0.3098   0.3341   31.8279   15.1623 5.2263 2.4897
         0.96   0.97   0.06820   0.3100   0.3338   32.9585   15.7181 5.0518 2.4093
         0.97   0.98   0.06340   0.3101   0.3334   34.0986   16.2805 4.8774 2.3287
         0.98   0.99   0.05870   0.3103   0.3330   35.2450   16.8481 4.7029 2.2481
         0.99   1.0    0.05410   0.3105   0.3326   36.3939   17.4194 4.5284 2.1675


     The table below is computed using the subconvexity bound a1 = 0.618, see Remark
     1.5(2)
           σ1   σ2       α       δ       d       C1       c˜1      C2        c˜2
          0.60 0.70 0.2167 0.3117 0.3434 7.9115 3.6668 11.3303 5.2513
          0.70 0.80 0.1595 0.3117 0.3418 13.8214 6.4369 9.5869 4.4648
          0.80 0.81 0.1539 0.3118 0.3416 14.5818 6.7949 7.8444 3.6554
          0.81 0.82 0.1483 0.3118 0.3414 15.3770 7.1697 7.6700 3.5762
          0.82 0.83 0.1427 0.3118 0.3412 16.2078 7.5617 7.4957 3.4971
          0.83 0.84 0.1371 0.3118 0.3410 17.0751 7.9713 7.3213 3.4179
          0.84 0.85 0.1315 0.3119 0.3407 17.9796 8.3991 7.1469 3.3387
          0.85 0.86 0.1260 0.3119 0.3405 18.9219 8.8453 6.9725 3.2594
          0.86 0.87 0.1205 0.3119 0.3403 19.9027 9.3103 6.7982 3.1801
          0.87 0.88 0.1150 0.3120 0.3400 20.9223 9.7945 6.6238 3.1008
          0.88 0.89 0.1096 0.3120 0.3398 21.9809 10.2980 6.4494 3.0215
          0.89 0.90 0.1042 0.3121 0.3395 23.0788 10.8209 6.2750 2.9422
          0.90 0.91 0.09890 0.3121 0.3392 24.2157 11.3635 6.1006 2.8628
          0.91 0.92 0.09360 0.3122 0.3389 25.3914 11.9256 5.9261 2.7833
          0.92 0.93 0.08840 0.3123 0.3386 26.6053 12.5071 5.7517 2.7039
          0.93 0.94 0.08320 0.3124 0.3383 27.8566 13.1078 5.5773 2.6244
          0.94 0.95 0.07810 0.3125 0.3379 29.1440 13.7273 5.4028 2.5448
          0.95 0.96 0.07310 0.3126 0.3376 30.4660 14.3650 5.2284 2.4653
          0.96 0.97 0.06820 0.3128 0.3372 31.8207 15.0203 5.0539 2.3856
          0.97 0.98 0.06340 0.3129 0.3368 33.2059 15.6924 4.8794 2.3059
          0.98 0.99 0.05870 0.3131 0.3364 34.6187 16.3800 4.7049 2.2262
          0.99 1.0 0.05420 0.3133 0.3360 36.0559 17.0819 4.5304 2.1464
