Pith. sign in

REVIEW 3 major objections 4 minor 35 references

Bounds for Mertens Sums

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper establishes explicit exponential-error bounds for the three classical Mertens sums for all x ≥ 2, more than doubling the previous decay constant and supplying corrected tables for the Mertens products.

desk verdict Useful explicit bounds for Mertens sums, but the exponential decay constant 0.8746 in Theorems 3 and 4 is unsupported by the proofs, which only deliver 0.84768. read the letter →

arxiv 2608.01498 v1 pith:HUJ7LBFK submitted 2026-08-02 math.NT

Samuel Broadbent , Andrew Fiori , Habiba Kadiri , Nathan Ng , Kirsten Wilk This is my paper · ORCID
classification math.NT MSC 11N0511M0611M26
keywords MertenssumsproductsexplicitboundsRiemannzetazeroszero-densityestimatesformulaprimenumbertheoremChebyshevfunctions
topics The Riemann HypothesisThe Twin Prime Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The classical 1874 theorems for prime sums state that $\sum_{p \le x} 1/p = \log \log x + M + o(1)$, with parallel statements for weighted prime-power sums and the Mertens products. This paper supplies explicit error terms in two shapes: an exponential form $A(x_0)(\log x)^{B} \exp(-C \sqrt{\log x})$ valid for all $x\ge x_0$, and a log form $A_\ell(x_0)/(\log x)^\ell$. The headline result is that for $\lambda(x)=\sum_{p\le x} 1/p$ the exponential bound holds with $A_\lambda(2)=9.2203$ and $C=0.84768$, more than doubling the decay constant of the previous record and moving the validity threshold from $x\ge e^{4635}$ down to $x\ge 2$. Analogous bounds are proved for $\Upsilon(x)$ and $\tilde\psi(x)$, and for the two Mertens products. The proofs run through a Riemann--Guinand explicit formula for $\lambda(x)$ and sharp bounds on weighted sums of zeta zeros.

What carries the argument

The carrying objects are the weighted zero sums $J_m(x)=\sum_{\rho} x^{\beta-1}/|\gamma|^{m+1}$, over nontrivial zeros $\rho=\beta+i\gamma$ of the Riemann zeta function. The paper bounds $J_1$ and $J_2$ by splitting the zeros into three ranges: low zeros with $|\gamma|<H$, where the Riemann hypothesis has been verified up to $H=3\cdot10^{12}$; zeros in the zero-free region, controlled by the constant $R=5.558691$; and zeros with real part above $5/8$, controlled by an explicit zero-density estimate $N(\sigma,T)\le c_1 T^{p(\sigma)}(\log T)^{q(\sigma)}+c_2(\log T)^2$. A telescoping argument converts this zero-density estimate into exponentially decaying bounds on $J_m(x)$, and a Riemann--Guin

What would settle it

Evaluate the claimed inequality (14) directly at every prime $p\le10^{10}$: a single prime with $|\lambda(p)-\log\log p-M|>9.2203\,(\log p)^{1/2}\exp(-0.84768\sqrt{\log p})$ would falsify Theorem 2(i). Alternatively, recompute $N(\sigma,T)$ by the method of [17] for $\sigma$ just above $5/8$ and for large $T$, and check whether the tabulated constants in [12, Table 7] are actually satisfied for all $\sigma\in(5/8,1)$; a single violation collapses the zero-sum bounds.

Watch

Extended reading notes

Core claim

The central claim is that the three Mertens sums satisfy explicit inequalities with the same shape as the best current Chebyshev-function bounds. In particular, Theorem 2 proves $|\lambda(x)-\log\log x-M|\le A_\lambda(x_0)(\log x)^{1/2}\exp(-C\sqrt{\log x})$ for every $x\ge x_0$, with $A_\lambda(2)=9.2203$ and $C=0.84768$, and also proves $|\lambda(x)-\log\log x-M|\le A_\ell(x_0)/(\log x)^\ell$ for $\ell=1,\dots,5$. Theorems 3 and 4 give matching bounds for $\Upsilon(x)=\sum_{p\le x}(\log p)/p$ and $\tilde\psi(x)=\sum_{n\le x}\Lambda(n)/n$, with $A_\Upsilon(2)=A_{\tilde\psi}(2)=9.2203$. From these bounds the paper derives two-sided inequalities for the Mertens products $\prod_{p\le x}(1-1/p)

Load-bearing premise

The whole exponential improvement rests on a previously tabulated bound for how many zeta zeros can deviate to the right of the critical line; if that bound is valid on a narrower range than claimed, or its constants are slightly too small, the factor-two gain collapses.

Editorial extensions

If this is right

  • For every $x\ge2$, the sum of reciprocal primes is pinned within $9.2203\,(\log x)^{1/2}\exp(-0.84768\sqrt{\log x})$ of $\log\log x+M$, so explicit estimates at any scale no longer require assuming the Riemann hypothesis.
  • The exponential decay constant for $\lambda(x)$ improves from $0.4183$ to $0.84768$, more than doubling the previous rate, and the range of validity drops from $x\ge e^{4635}$ to all $x\ge2$.
  • The same exponential-form bounds now hold for $\sum_{p\le x}(\log p)/p$ and $\sum_{n\le x}\Lambda(n)/n$, with constant $9.2203$ at $x_0=2$, together with log-form bounds at five different powers of $\log x$.
  • The Mertens products $e^{-\gamma}/\log x\,\prod_{p\le x}(1-1/p)$ and $e^{\gamma}\log x\,\prod_{p\le x}p/(p-1)$ get explicit two-sided inequalities with error $A_\ell(x_0)/(\log x)^\ell$ for $\ell=1,\dots,5$, recovering results previously derived from an invalid argument in a 2018 article.
  • The zero-sum bounds $J_m(x)$ are stated with general parameters, so future improvements in the zero-free region, verification height, or zero-density estimate can be substituted without reworking the proof.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the $J_m(x)$ bounds feed into the error term only through the explicit formula, the same machinery should apply to other prime-weighted averages such as $\sum_{p\le x}(\log p)^k/p$ for $k>1$, though the paper does not state such consequences.
  • The new exact explicit formula (31) could be studied on its own: refining the incomplete-gamma terms or the $\psi-\vartheta$ correction may further sharpen the constant $A_\lambda$, an avenue the paper leaves open.
  • If the zero-density estimate were valid closer to the critical line, the same proof would automatically yield bounds at smaller $x_0$; any future increase in the verified height $H$ would immediately reduce the $a_m$ and $b_m$ parts of $J_m(x)$.
  • The conjectured true size of the error, on the order of $\sqrt{x}(\log\log\log x)^2$, remains far below the paper's exponential bounds, so the contribution is to explicit certainty rather than to the conjectured asymptotics.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops explicit bounds for the Mertens sums λ(x)=∑_{p≤x}1/p, Υ(x)=∑_{p≤x}(log p)/p, and eψ(x)=∑_{n≤x}Λ(n)/n, together with corollaries for the Mertens products. The main ingredients are: a new Riemann–Guinand type explicit formula for λ(x), bounds for weighted zero sums J_m(x)=∑_ρ x^{β-1}/|γ|^{m+1}, the classical zero-free region with R=5.558691, the partial RH verification height H=3·10^{12}, and a zero-density estimate from Kadiri–Lumley–Ng as tabulated in [12]. Theorems 2–4 give exponential-form and log-form bounds with explicit constants, and Theorem 7 gives bounds for J_m(x). The paper also claims to recover and correct Dusart's results on Mertens sums and products and to improve Vanlalngaia's exponential decay constant by a factor greater than 2.

Significance. If correct, the paper supplies the first corrected post-Dusart tables for these sums and products, extends the validity range to all x≥2, and gives a substantial improvement in the exponential decay rate over Vanlalngaia. The explicit formulae and the systematic treatment of J_m(x) are likely to be useful for future improvements and for other problems in explicit number theory. The strength of the paper is that all constants are given as explicit functions of the published inputs (R, H, the zero-density constants, and the ψ/θ bounds), and the framework is designed to be updated when those inputs improve. However, a central stated improvement—the decay constant 0.8746 in Theorems 3(i) and 4(i)—is not supported by the proof, which only yields C=0.84768. This overstatement affects the headline theorems, the tables, and the comparison with prior work. The underlying method appears sound, but the stated constants must be corrected before the paper can be accepted.

major comments (3)
  1. [Theorems 3(i), 4(i) and §4, eqs. (17), (20), (152), (161); Tables 11–12; Remark 1] The stated decay constant 0.8746 is not derived anywhere. The proof of Theorem 3 uses φ(x)=9.2204(log x)^{3/2} exp(-C√log x) with C=0.84768 and combines A_ϑ(x0) exp(-C√log x) with D'(x0) exp(-C√2√log x); since exp((0.8746-0.84768)√log x) is unbounded, no finite constant A_Υ can convert the proven bound into one with decay 0.8746. The same applies to Theorem 4. The value 0.8746 appears in the theorem statements, Tables 11–12, and Remark 1 (including eq. (23) for λ, where Theorem 2 uses C=0.84768). Please either replace 0.8746 by C=0.84768 throughout and recompute the quoted comparisons, or prove that a zero-free region with R=(2/0.8746)^2 is being used and propagate it through the whole argument.
  2. [Definition 11, eqs. (48)–(49); Proposition 26; Theorem 7] All exponential improvements pass through the zero-density estimate (ZDB), which is cited from [12, Table 7] rather than re-derived. The paper assumes the estimate holds for every σ>5/8 with p(σ)<1 and uses the tabulated c1,c2,p,q. Since a small change in p(σ) or a narrowing of its validity range would collapse the factor-2 gain in C√(m+1), the manuscript should state explicitly the provenance of the table, whether the constants are uniform on each σ-interval, and how sensitive the final constants A_m and A_λ, A_Υ, A_eψ are to small perturbations of the tabulated p(σ). A concrete check would be to verify, from the raw KLN data, that p(σ)<1 for every σ in (5/8,1) with the printed constants; otherwise the relevant theorems should state the restricted range of σ where the bound is certified.
  3. [Proof of Theorem 3, eq. (151)] The piecewise definition of D'(x0) writes x0∈[σ_j,σ_{j+1}], but the intervals should be over the y_j=exp(t(1,σ_j)) thresholds, as is done correctly for D''(x0) in eq. (160) of Theorem 4. As written, (151) is not meaningful because σ_j are numbers in (0.65,0.9), not endpoints of the x0-range. This should be corrected to [y_j,y_{j+1}].
minor comments (4)
  1. [Table 10 vs Theorem 2(i); Table 12 vs Theorem 4(i)] The exponents in the table headers are inconsistent with the theorem statements. Table 10 writes (log x)^{3/2} for λ, while Theorem 2(i) and eq. (14) use (log x)^{1/2}. Table 12 writes (log x)^{1/2} for eψ, while Theorem 4(i) and eq. (20) use (log x)^{3/2}.
  2. [Abstract and §1] The abstract contains the typo 'weighted sums of zeros of zeros of the zeta function'.
  3. [Table 1] The header lists A_ψ(x0)/A_ϑ(x0)/A_λ(x0)/A_Υ(x0)/A_eψ(x0) but only one numerical column is shown. If the values coincide at the displayed precision, this should be stated explicitly; otherwise separate columns are needed.
  4. [Remark 1, eq. (23)] The displayed comparison for Theorem 2 uses 0.8746, but Theorem 2(i) and eq. (14) use C=0.84768. This is part of the same inconsistency as the major comment, but it should also be fixed in the comparison table/formula.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central bounds are derived from explicit cited inputs (zero-free region, zero-density estimate, and prior psi/theta bounds), with no fitted parameter renamed as a prediction. The paper's heavy reliance on same-author-group citations is mutual dependence, not circularity. The 0.8746-vs-0.84768 constant discrepancy is an internal correctness issue, not a circular step

full rationale

The derivation chain is genuinely constructive rather than circular. Theorem 2(i) defines A_lambda(x0) explicitly in (74) as A_theta(x0) + A''(x0) exp(-C(sqrt(2)-1) sqrt(log x0)), where A_theta comes from Theorem 1 (cited from [12],[13]) and A''(x0) is assembled from bounds on the zero sums J_1, J_2 in Theorem 7. Theorem 7 itself is proved from Definition 11 (ZDB), which is a cited zero-density estimate from Kadiri-Lumley-Ng [17], plus the classical zero-free region R from [22] and the partial RH verification H from [28]. No step fits a parameter to the Mertens sums being bounded; the small-x verification in the proof of Theorem 2 uses the Rosser-Schoenfeld bounds as a sanity check, not as an input to the constants. Likewise Theorems 3 and 4 use Proposition 28 and explicit formulas, with A_Upsilon(x0) = A_theta(x0) + D'(x0) exp(-C(sqrt(2)-1) sqrt(log x0)) and A_psitilde(x0) = A_psi(x0) + D''(x0) exp(-C(sqrt(2)-1) sqrt(log x0)). These are explicit functions of cited inputs, not of the target error terms. The cited results [4],[12],[13],[17] share authors with this paper, but they are independently published, parameter-free results with stated assumptions that do not include the present theorems; under the rubric, that is real support, not circularity. The one notable anomaly—Theorems 3(i) and 4(i) and Remark 1 printing 0.8746 while the proof and eq. (12) use C = 0.84768—does not exhibit a reduction of the conclusion to an input; it is an apparent internal inconsistency or unsupported stronger constant, which is a correctness risk rather than a circularity. Therefore no circular step is established by quotation, and the circularity score is low.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

Seven external inputs are carried from prior literature: zero-free region, zero-density estimate, partial RH verification, explicit psi/theta bounds, the corrected Ramar e-Saouter lemma, the Rosser-Schoenfeld summation identities, and the Buethe-based eta tables. Two hand-chosen computational parameters (sigma-grid and K) and one interval partition enter the stated constants; none are fitted to data. No invented entities are introduced: J_m(x), the incomplete gamma functions, and formula (31) are standard objects or rearrangements of known ingredients.

free parameters (3)
  • sigma-grid for piecewise constants = 0.65, 0.70, 0.80, 0.90
    Hand-chosen values in (72), (151), (160) defining A''(x0), D'(x0), D''(x0). The theorems are rigorous for any sigma0 > 5/8, but the stated numeric constants inherit this choice.
  • K, partition size in G_m(x0, K) = 10000
    Used in Proposition 26 to define C_m(x0, sigma0) and hence A_m in (123); larger K refines the bound, and the final table values depend on it.
  • interval partition for log-form J_m bounds = intervals [x0 e^j, x0 e^{j+1}], sigma0 = 0.9, sigma1 = 0.99
    In Theorem 7(ii) and (130), delta-tilde_{m,ell} is computed over a partition of [x0, x1]; the chosen intervals and sigma values affect the delta values used in Tables 4-9.
assumptions (7)
  • domain assumption Explicit zero-free region: zeta(s) nonvanishing for Re(s) >= 1 - 1/(R log|Im s|), |Im s| >= 3, with R = 5.558691
    Equations (46)-(47), cited to Mossinghoff-Trudgian-Yang [22]. Used throughout Section 3 to control x^{beta-1} inside zero sums.
  • domain assumption Zero-density bound (ZDB) of Kadiri-Lumley-Ng: N(sigma,T) <= c1(sigma) T^{p(sigma)} (log T)^{q(sigma)} + c2(sigma)(log T)^2 for sigma > sigma0 with sigma0 > 5/8 and 0 < p(sigma) < 1
    Definition 11, equations (48)-(49), constants as tabulated in [12, Table 7]. This is the load-bearing input for the J_m(x) bounds in Section 3 and explains the 5/8 threshold throughout.
  • domain assumption Partial RH up to H = 3,000,175,332,800: every zero with 0 < gamma < H has beta = 1/2
    Equations (50)-(51), cited to Platt-Trudgian [28]. Used to cancel all |gamma| < H contributions in Lemma 22 and Proposition 26.
  • domain assumption Explicit bounds for psi(x) and theta(x) of Theorem 1, with A_psi(2) = A_theta(2) = 9.22022 and eta_ell, eta-tilde_ell from Tables 2-3
    Equations (9)-(13), cited to [4], [12], [13]. Every Mertens bound in Theorems 2-4 starts from |theta-x|/x or |psi-x|/x, so the final constants inherit these inputs.
  • domain assumption Ramar e-Saouter Lemma 4 explicit formula for integrals of (psi(t)-t)g(t), with the sign of log(2pi) corrected to negative
    Lemma 20 and Proposition 13: the paper asserts [30] contains a sign error and relies on the corrected version to derive (55). If the sign correction is wrong, the explicit formula changes sign and structure.
  • domain assumption Rosser-Schoenfeld partial summation identities (34) for lambda(x) and (137) for Upsilon(x), plus their small-x bounds used in the range x <= e^464.3
    Opening of Section 2 and proofs of Theorems 2-4; cited to [32]. The two-pronged proof of Theorem 2(i) depends on these classical bounds for small x.
  • domain assumption Validity of Buethe's method as corrected in Bhattacharjee [3] and of the FKS method for the computed eta_ell(x0), eta-tilde_ell(x0) in Tables 2-3
    Remark after Theorem 1: the small-x and large-x bounds for psi and theta that feed all log-form results rest on these external computations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bounds for Mertens Sums." pith.science (2026). https://pith.science/paper/HUJ7LBFK

@misc{pith2026260801498,
  author       = {Pith},
  title        = {Pith review of: Bounds for Mertens Sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HUJ7LBFK}},
  note         = {Machine review of arXiv:2608.01498}
}
abstract

In this article we provide new bounds for the Mertens sums and products including $\sum_{p \le x} p^{-1}$ and $\prod_{p \le x} (1-\frac{1}{p})$ which provide superior exponential and log type bounds for these sums in all ranges. These weighted prime number sums and products were extensively studied by Rosser and Schoenfeld (1962) and are employed in a wide range of applications in number theory, cryptography, and combinatorics. Extensive tables are provided in this article which will be useful for these types of applications. The main new ideas in this article are sharp bounds for weighted sums of zeros of zeros of the zeta function. We make use of a novel technique of Fiori-Kadiri-Swidinsky (2023) which relies on a recent explicit zero-density estimate for $N(\sigma,T)$ of Kadiri-Lumley-Ng (2018). The bounds and techniques in this article for weighted zeros sums will likely be useful in many other arithmetic applications. Our main theorem significantly improves the exponential decay result of Vanlalngaia (2017) and fills a gap in the literature by correcting work of Dusart (2018). The results are also presented in a way that are amenable to future improvements. In addition, we prove an exact ``Riemann-Guinand explicit formula" for the Mertens sum $\sum_{p \le x} p^{-1}$ that appears to be new.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 31 canonical work pages

  1. [12]

    Fiori, H

    A. Fiori, H. Kadiri, and J. Swidinsky. Sharper bounds for the Chebyshev functionψ(x).J. Math. Anal. Appl., 527(2):Paper No. 127426, 28, 2023

    work page 2023

  2. [1]

    C. Axler. New estimates for some functions defined over primes.Integers, 18:Paper No. A52, 21, 2018

    work page 2018

  3. [2]

    Bellotti, T

    C. Bellotti, T. Trudgian, and A. Yang. Zero-free regions inspired by work of heath-brown, 2026

    work page 2026

  4. [3]

    Bhattacharjee

    S. Bhattacharjee. A survey of b¨ uthe’s method for estimating prime counting functions. M.sc. thesis, University of Lethbridge, Lethbridge, Alberta, Canada, 2024. Department of Mathematics and Computer Science

    work page 2024

  5. [4]

    Broadbent, H

    S. Broadbent, H. Kadiri, A. Lumley, N. Ng, and K. Wilk. Sharper bounds for the Chebyshev functionθ(x). Math. Comp., 90(331):2281–2315, 2021

    work page 2021

  6. [5]

    J. B¨ uthe. Estimatingπ(x) and related functions under partial RH assumptions.Math. Comp., 85(301):2483–2498, 2016

    work page 2016

  7. [6]

    Chirre and H

    A. Chirre and H. A. Helfgott. Optimal bounds for sums of non-negative arithmetic functions, 2025

    work page 2025

  8. [7]

    Davenport.Multiplicative number theory, volume 74 ofGraduate Texts in Mathematics

    H. Davenport.Multiplicative number theory, volume 74 ofGraduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2000. Revised and with a preface by Hugh L. Montgomery

    2000

Show all 35 references
  1. [8]

    P. Dusart. In´ egalit´ es explicites pourψ(X),θ(X),π(X) et les nombres premiers.C. R. Math. Acad. Sci. Soc. R. Can., 21(2):53–59, 1999

    1999

  2. [9]

    P. Dusart. Estimates ofψ, θfor large values ofxwithout the Riemann hypothesis.Math. Comp., 85(298):875–888, 2016

    2016

  3. [10]

    P. Dusart. Explicit estimates of some functions over primes.Ramanujan J., 45(1):227–251, 2018

    2018

  4. [11]

    Faber and H

    L. Faber and H. Kadiri. New bounds forψ(x).Math. Comp., 84(293):1339–1357, 2015

    2015

  5. [13]

    Fiori, H

    A. Fiori, H. Kadiri, and J. Swidinsky. Sharper bounds for the error term in the prime number theorem.Res. Number Theory, 9(3):Paper No. 63, 19, 2023

    2023

  6. [14]

    Integrated explicit analytic number theory network.https://www.ipam.ucla.edu/news-research/special-projects/ integrated-explicit-analytic-number-theory-network/

    Integrated Explicit Analytic Number Theory Network. Integrated explicit analytic number theory network.https://www.ipam.ucla.edu/news-research/special-projects/ integrated-explicit-analytic-number-theory-network/. Accessed: 2026-04-28

    2026

  7. [15]

    D. R. Johnston. Improving bounds on prime counting functions by partial verification of the Riemann hypothesis. Ramanujan J., 59(4):1307–1321, 2022. 36

    2022

  8. [16]

    D. R. Johnston and A. Yang. Some explicit estimates for the error term in the prime number theorem.J. Math. Anal. Appl., 527(2):Paper No. 127460, 23, 2023

    2023

  9. [17]

    Kadiri, A

    H. Kadiri, A. Lumley, and N. Ng. Explicit zero density for the Riemann zeta function.J. Math. Anal. Appl., 465(1):22–46, 2018

    2018

  10. [18]

    Lamzouri

    Y. Lamzouri. A bias in Mertens’ product formula.Int. J. Number Theory, 12(1):97–109, 2016

    2016

  11. [19]

    Landau.Handbuch der Lehre von der Verteilung der Primzahlen

    E. Landau.Handbuch der Lehre von der Verteilung der Primzahlen. 2 B¨ ande. Chelsea Publishing Co., New York,

  12. [20]

    F. Mertens. Ein beitrag zur analytischen zahlentheorie.J. Reine Angew. Math., 78:46–63, 1874

  13. [21]

    H. L. Montgomery. The zeta function and prime numbers. InProceedings of the Queen ’s Number Theory Confer- ence, 1979 (Kingston, Ont., 1979), volume 54 ofQueen ’s Papers in Pure and Appl. Math., pages 1–31. Queen’s Univ., Kingston, ON, 1980

    1979

  14. [22]

    M. J. Mossinghoff, T. S. Trudgian, and A. Yang. Explicit zero-free regions for the Riemann zeta-function.Res. Number Theory, 10(1):11, 2024

    2024

  15. [23]

    N. Ng. Prime number error terms, 2025

    2025

  16. [24]

    F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors.NIST Handbook of Mathematical Functions. Cambridge University Press, 2010

    2010

  17. [25]

    I. Pinelis. Exact lower and upper bounds on the incomplete gamma function.Math. Inequal. Appl., 23(4):1261– 1278, 2020

    2020

  18. [26]

    J. Pintz. On the remainder term of the prime number formula. II. On a theorem of Ingham.Acta Arith., 37:209– 220, 1980

    1980

  19. [27]

    Platt and T

    D. Platt and T. Trudgian. The error term in the prime number theorem.Mathematics of Computation, 90(328):871–881, 2021

    2021

  20. [28]

    Platt and T

    D. Platt and T. Trudgian. The Riemann hypothesis is true up to 3·10 12.Bull. Lond. Math. Soc., 53(3):792–797, 2021

    2021

  21. [29]

    Ramar´ e

    O. Ramar´ e. Explicit estimates for the summatory function of Λ(n)/nfrom the one of Λ(n).Acta Arith., 159(2):113–122, 2013

    2013

  22. [30]

    Ramar´ e and Y

    O. Ramar´ e and Y. Saouter. Short effective intervals containing primes.J. Number Theory, 98(1):10–33, 2003

    2003

  23. [31]

    Ramar´ e and S

    O. Ramar´ e and S. Z. Alterman. M¨ obius function and primes: an identity factory with applications, 2023

    2023

  24. [32]

    J. B. Rosser and L. Schoenfeld. Approximate formulas for some functions of prime numbers.Illinois J. Math., 6:64–94, 1962

    1962

  25. [33]

    Schoenfeld

    L. Schoenfeld. Sharper bounds for the Chebyshev functionsθ(x) andψ(x). II.Math. Comp., 30(134):337–360, 1976

    1976

  26. [34]

    Vanlalngaia

    R. Vanlalngaia. Explicit Mertens sums.Integers, 17:Paper No. A11, 18, 2017. 37 AppendixA.Tables of values Table 1.Values forA ψ(x0),A ϑ(x0),A λ(x0),A Υ(x0),A eψ(x0). b= log(x 0) Aψ(x0)/Aϑ(x0)/A λ(x0)/A Υ(x0)/A eψ(x0) 2 9.2203 200 000 9.235416396 300 000 8.962915751 400 000 8.7...

    2017

  27. [1953]

    2d ed, With an appendix by Paul T. Bateman

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.