New bounds in R.S. Lehman's estimates for the difference $\pi\left( x\right) -li\left( x\right) $

Michael Revers

REVIEW 3 major objections 3 minor 17 references

New bounds in R.S. Lehman's estimates for the difference $\pi\left( x\right) -li\left( x\right) $

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves sharper error bounds for locating sign changes of π(x) − li(x), removing a lower-bound restriction on the window radius and shrinking certified crossover intervals near 10^316.

desk verdict A solid, incremental improvement to Lehman's bounds with a credible main theorem, but missing proof for Theorem 7.1 and thin numerical reproducibility; the alleged R6 algebra error appears to be a misreading of the typesetting. read the letter →

arxiv 2501.04488 v2 pith:6MP5SON7 submitted 2025-01-08 math.NT

Michael Revers This is my paper · ORCID

classification math.NT MSC 11M2611N0511Y1111Y35

keywords PrimecountingfunctionSkewesnumbersRiemannzerosCrossoversLogarithmicintegralExplicitformulaErrorboundsπ(x)−li(x)signchanges

topics The Riemann Hypothesis The 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

This paper sharpens the error estimates in a 1966 integral method for locating sign changes of $\pi(x)-\mathrm{li}(x)$, the difference between the prime-counting function and the logarithmic integral. The main theorem improves the earlier error terms, most notably replacing the old $S_6/S_3$ term with a new $R_6$ term that is smaller by an exponential factor and eliminating the lower restriction $2A/\alpha\le\eta$ entirely; the only remaining condition on the window radius is $\eta<\omega/100$. The added $1/(\omega\rho^2)$ term in the explicit-formula sum induces better $R_2$ and $R_4$ error terms. Because these error terms control how narrow a certified interval can be, the improvement directly translates into smaller search intervals: the certified window near $10^{316}$ shrinks by roughly a third, and a run of more than $4.6\times10^{154}$ consecutive integers where $\pi(x)>\mathrm{li}(x)$ is established near $e^{727.952018}$. The paper also tabulates heuristic candidates for earlier crossovers below $10^{316}$.

What carries the argument

The load-bearing object is the integrated explicit formula. Instead of checking $\pi(x)-\mathrm{li}(x)$ pointwise, the paper studies $I(\omega,\eta)$, a Gaussian-weighted average of the scaled difference $u e^{-u/2}(\pi(e^u)-\mathrm{li}(e^u))$ over a logarithmic interval of radius $\eta$ around $\omega$. Positivity of $I(\omega,\eta)$ forces $\pi(x)>\mathrm{li}(x)$ somewhere in $[e^{\omega-\eta},e^{\omega+\eta}]$. The argument expands $\mathrm{li}(e^{u\rho})$ to second order, producing the extra $1/(\omega\rho^2)$ sum; this is what creates the improved $R_2$ and $R_4$ terms. The $R_6$ bound comes from repeated partial integration of the analytic function $f_\rho(z)=\rho z e^{-\rho z}\mathrm{li}(e^{\rho z})e^{-\alpha(z-\omega)^2/2}$ over a sector, with derivatives controlled by contour estimates at radius $r=\sqrt{n/\alpha}$ and truncation $N=[A^2/\alpha]$, followed by summing the high zeros $|\gamma|>A$ with the zero-counting estimate in Lemma 4.7(a).

What would settle it

Recompute the numerical sums $S_1^*$ and $S_2^*$ in Section 6 with an independent, higher-precision set of zeta zeros: if the discrepancy with the paper's values exceeds the estimated $\Delta S_1+\Delta S_2$ (about $1.9\times10^{-5}$ for the first window), the positive lower estimates for $I(\omega,\eta)$ in Tables 3 and 4 can turn negative. A second, more drastic falsifier would be finding a zeta zero with $\gamma\le A$ that lies off the critical line.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 3.1. With $K(x)=\sqrt{\alpha/2\pi}\,e^{-\alpha x^2/2}$ and $I(\omega,\eta)=\int_{\omega-\eta}^{\omega+\eta}K(u-\omega)\,u e^{-u/2}(\pi(e^u)-\mathrm{li}(e^u))\,du$, the theorem asserts that if all zeta zeros with imaginary part up to height $A$ lie on the critical line, and if $5A/(4\omega)\le\alpha\le A^2$ and $0<\eta<\omega/100$, then for $2\pi e\le T\le A$, $$I(\omega,\eta)\ge -1-\sum_{0<|\gamma|\le T}$e^{{i\omega\gamma}}$\left(\frac{1}{\rho}+\frac{1}{\omega\$rho^{2}$}\right)$e^{{-\gamma^2/2\alpha}}$+R,$$ with $|R|\le R_1+\cdots+R_6$. The new $R_6$ term, $\left(1+\frac{22}{A\omega}\right)A\log A\left(\frac{8.283}{A}e^{-\frac{\alpha}{4}\eta^2+\frac{\omega+\eta}{2}}+7.152\,\eta e^{-\frac{A^2}{2\alpha}+\frac{\omega+\eta}{2}}\right)$, improves the older $S_6$ term because its leading exponential has $\alpha/4$ rather than $\alpha/8$ in the exponent after substituting the old lower bound for $\eta$, and because the large factor $A$ cancels in the first exponential; it also makes the lower condition on $\eta$ unnecessary. If the Riemann hypothesis holds in full, $R_6$ and the parameter conditions are omitted.

Load-bearing premise

The proof assumes the Riemann hypothesis is verified up to height $A$, meaning every zeta zero with imaginary part $\gamma\le A$ lies on the critical line; if one such zero lies off the line, the sharp $R_6$ term must be replaced by a weaker expression and the certified numerical windows may no longer hold.

Editorial extensions

If this is right

The 2010 certified window at $\omega\approx727.952018$ can be resized from $\eta=0.00016$ to $\eta=0.0001061$ while keeping the lower estimate for $I(\omega,\eta)$ positive, a reduction of about $33.68\%$.
The 2010 interval near $\omega\approx727.95134$ can be resized from $\eta=0.000022833$ to $\eta=0.0000159$, a reduction of about $30.80\%$.
In the vicinity of $e^{727.952018}$, more than $4.61877\times10^{154}$ successive integers satisfy $\pi(x)>\mathrm{li}(x)$ (Theorem 6.1).
The double-summation term $S_2(\alpha,\omega,T)$ is bounded in magnitude by about $1/(21\omega)$, so the extra computational cost is small while it enables the sharper $R_2$, $R_4$, and $R_5$ error terms.
The same contour and partial-integration strategy yields an analogous sharper $R_5$ error term for the alternative kernel studied in Section 7.

Reading between the lines

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

Iterating the expansion of $\mathrm{li}(e^{u\rho})$ beyond second order, to $1/\rho^3$ terms, could shrink $R_2$ and $R_4$ further at the cost of heavier summation; the paper leaves this open.
With the lower bound on $\eta$ removed, $\eta$ becomes a free optimization variable, so a systematic scan over $(\alpha,\eta,T)$ could shrink the certified windows further than the resizing shown in Tables 3 and 4.
The candidates below $10^{316}$ in Section 8 are heuristic sums $F_T(\omega)$ rather than certified bounds; a natural next step would be to run Theorem 3.1 at each listed $\omega$ to test whether any can be certified or ruled out.
Because the theorem only needs the Riemann hypothesis up to height $A$, any future extension of verified zero computations to larger heights automatically improves the numerical conclusions without changing the proof.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick read on Revers's paper. The new content is genuine: an improved R6 term, removal of the lower eta condition, and an adapted R5 for the Saouter-Trudgian-Demichel kernel. The numerical tables show real reductions in eta (about 33% and 30%) and Theorem 6.1 gives a run of consecutive integers. The paper builds on the double-summation technique already in the 2015 paper, so the novelty is incremental, but it is a legitimate extension.

The main theorem looks plausible. The one algebraic concern in the stress-test note—an extra factor of alpha in R6—does not hold up. The note reads (5.18) as eA√alpha; the correct bound is eA/√alpha (the printed typesetting is easy to misread). With that, the constants 8.283 and 7.152 come out right and the alpha cancels. So I would not sink the paper on that.

Soft spots are real but not fatal. Theorem 7.1 is stated and used without proof; the author says it follows by standard arguments, but for a theorem that carries numerical weight, that is thin. The interchange of summation and integration in (5.1) is waved through with an appeal to Arzela's theorem; since the zero series is only conditionally convergent, a referee should ask for the details. There is no reproducible code or data release; the numerical claims rest on Mathematica runs and an assumed 10^-9 zero accuracy. The paper also has a number of typos (w vs omega, missing parentheses) that make the proof harder to check.

In sum: this is a serious, honest piece of work within a well-studied program. It does not solve the first-crossover problem, and it does not need to. The improvements are modest but real. If I were the editor, I would send it to a referee who can check the analysis in Section 5 and push for a proof of Theorem 7.1 and the interchange step. I would not desk-reject it.

Referee Report

3 major / 3 minor

Summary. The paper claims improved effective error bounds in the integrated Lehman formula for π(x)−li(x). Theorem 3.1 introduces new error terms R1 through R6 and removes the lower bound on η, replacing it with 0<η<ω/100. The derivation is sketched in Section 5; the author then applies the theorem to refine known crossover regions near 10^316 and discusses numerical candidates for earlier crossovers. Section 7 states an analogous refinement for the Saouter–Trudgian–Demichel kernel.

Significance. If the main theorem is correct, the paper gives a nontrivial improvement over Lehman's S3 term and offers sharper certified intervals for known crossover regions. The numerical work uses explicit, previously certified parameters and reproducible zero computations. However, the proof of the central R6 term has a serious algebraic gap, and one additional theorem is stated without proof. The potential value of the result justifies a major revision, but the paper should not be accepted with the current proof.

major comments (3)

[Section 5.2, equations (5.17)–(5.20)] The derivation of R6 does not close algebraically. With the bound (5.18) as printed, the factor eA√α/(√e−1) inside the bracket, after multiplication by the prefactors in (5.17) and summation over |γ|>A via Lemma 4.7(a), produces a term proportional to α log A rather than log A. Even if (5.18) is corrected to eA/√α (which is what follows from n! ≤ (n/e)^n e√n), the n=0 term yields √α/A and the η-term yields Aη, not the 1/A and η that appear in (5.20). Thus the stated R6 is not a consequence of the written proof. This is load-bearing: the claimed improvement over Lehman's S3, and the numerical tables in Section 6, depend directly on this bound. Please re-derive R6 carefully and rework the numerical applications with the corrected constants.
[Section 5, equation (5.1)] The interchange of summation and integration for the infinite sum ∑_{|γ|>A} is justified only by an appeal to Arzela's bounded convergence theorem. This needs a rigorous check: the series over zeros is only conditionally convergent, and the sum of absolute values ∑ 1/γ diverges, so a dominated-convergence argument must explicitly show uniform convergence after the Gaussian factor and the integral are taken into account. The present one-sentence justification is insufficient for a formal proof.
[Section 7, Theorem 7.1] Theorem 7.1 is stated as a new result but the text says 'we do not present a detailed proof'. This is not acceptable for a formal paper: the theorem is used in Remark 3.7 to support an additional refinement, and the proof cannot be reduced to 'standard arguments' without at least a complete derivation of the analogous R5 term, including the constants 13.840 and 11.951. Please provide a full proof or a precise reference where this exact statement is proved.

minor comments (3)

[Theorem 3.1] There is a typographical inconsistency: the condition is written 'w−η≥44.22' in one place and 'ω−η≥44.22' elsewhere.
[Section 5.2, equation (5.20)] The numerical constants 8.283 and 7.152 need to be recomputed after the algebra of (5.17)–(5.19) is corrected; the displayed factor 4√(2π) does not match the preceding line.
[Section 6.4] The proof of the 'run of consecutive integers' should be more explicit about the switch from u to x: the bound is stated for u∈[ω−η,ω+η], but the subsequent variable y is measured from an unspecified point x; please clarify the quantifiers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new error bounds are derived from external analytic estimates and independent numerical zero data, not from the conclusion they support.

full rationale

No circular step is present. The core derivation in Sections 4 and 5 begins from the Riemann-von Mangoldt formula, Dusart's external prime-counting bounds, and standard lemmas on zeta-zero sums (Lemmas 4.1-4.8), then produces the explicit error terms R1 through R6 by integral estimates, partial integration, and summation over zeros. In particular, the R6 term in Section 5.2 is obtained by bounding the tail contribution from zeros with |gamma|>A through the analytic estimates (5.10)-(5.19) and Lemma 4.7(a); it does not assume the desired lower bound for I(omega,eta). The numerical section applies the theorem with published values of omega, alpha, T, and A, and then chooses eta to minimize the stated error terms; this is legitimate parameter optimization within a proven bound, not fitting a parameter to the predicted positivity. The only book result imported from outside is Plymen's Theorem 6.6 for R1, and that reference is not authored by the present author; there is no load-bearing self-citation chain. Even if the skeptic's algebraic complaint about the passage from (5.17)-(5.19) to (5.20) were correct, that would be a possible proof error or correctness risk, not a circular dependence. Since the theorem's error terms do not reduce by construction to the numerical positivity claims they support, the paper is not circular.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. It relies on standard explicit formulas, known bounds for π(x), and computational assumptions about zeta zeros. The free parameters are inherited from prior literature or numerically optimized, and are not fitted to force the conclusion.

free parameters (5)

eta = 0.0001061, 0.0000159 (optimized in Section 6)
Window half-width; optimized numerically to maximize the lower bound for I(ω,η).
alpha = 1.34e11, 6e12, 1.6e15 (from prior papers)
Gaussian kernel sharpness; inherited from Bays-Hudson, Saouter-Demichel and [14].
omega = 727.952018, 727.95134 (from prior papers)
Center of the search interval; taken from Chao-Plymen and Saouter-Demichel.
A = 1.022e7, 6.85e7, 1.13e9 (from prior papers)
Height up to which the Riemann hypothesis is assumed; based on earlier verifications.
T = 1131944.4718, 10379599.7274 (zero count dependent)
Truncation height for the zero sum; chosen from available zero computations.

assumptions (6)

domain assumption Riemann hypothesis up to height A: β = 1/2 for all zeros with 0 < γ ≤ A
Used throughout Section 5 to simplify the error terms and in the statement of Theorem 3.1.
standard math Riemann-von Mangoldt explicit formula for Π0(x)
Used in Section 4.1 to express π(x) - li(x) as a sum over zeta zeros.
standard math Dusart's explicit bounds for π(x) and related prime-counting estimates
Used in Section 4.1 (Theorem 5.1 of [6]) to bound the prime-power contribution.
standard math Backlund's explicit bound on the number of zeros N(T)
Used in Lemma 4.6 to estimate sums over zeros, following [10].
domain assumption Computed zeta zeros are accurate to within 10^-9 in imaginary part
Assumed in Section 6.1 to bound the numerical errors ΔS1 and ΔS2.
standard math Interchange of summation and integration in (5.1) is valid
Asserted via Arzela's bounded convergence theorem, but the zero series is conditionally convergent, so the application needs justification.

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Cite this review

Pith. "Pith review of New bounds in R.S. Lehman's estimates for the difference $\pi\left( x\right) -li\left( x\right) $." pith.science (2026). https://pith.science/paper/6MP5SON7

@misc{pith2026250104488,
  author       = {Pith},
  title        = {Pith review of: New bounds in R.S. Lehman's estimates for the difference $\pi\left( x\right) -li\left( x\right) $},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6MP5SON7}},
  note         = {Machine review of arXiv:2501.04488}
}

abstract

We denote by $\pi\left( x\right) $ the usual prime counting function and let $li\left( x\right) $ the logarithmic integral of $x$. In 1966, R.S. Lehman came up with a new approach and an effective method for finding an upper bound where it is assured that a sign change occurs for $\pi\left( x\right) -li\left( x\right) $ for some value $x$ not higher than this given bound. In this paper we provide further improvements on the error terms including an improvement upon Lehman's famous error term $S_{3}$ in his original paper. We are now able to eliminate the lower condition for the size-length $\eta$ completely. For further numerical computations this enables us to establish sharper results on the positions for the sign changes. We illustrate with some numerical computations on the lowest known crossover regions near $10^{316}$ and we discuss numerically on potential crossover regions below this value.

Figures

Figures reproduced from arXiv: 2501.04488 by the authors.

**Figure 2.** Regions 1020 to 10320, including the te Riele region. We will give some remarks. Remark 8.1. The computer plots in [PITH_FULL_IMAGE:figures/full_fig_p037_2.png] view at source ↗

**Figure 3.** Enlarged regions for possible crossovers, including the regions of [PITH_FULL_IMAGE:figures/full_fig_p040_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Abramowitz, I

M. Abramowitz, I. Stegun,Handbook of mathematical functions , Dover Publications, New York (1970),

work page 1970

[2] [2]

Backlund, Über die Nullstellen der Riemannschen Zetafunktion, Acta

R.J. Backlund, Über die Nullstellen der Riemannschen Zetafunktion, Acta. Math. 41 (1918), 345-375

work page 1918

[3] [3]

Bays and R.H

C. Bays and R.H. Hudson, A new bound for the smallestx withπ (x)> li (x), Math. Comp. 69 No. 231 (2000), 1285-1296

work page 2000

[4] [4]

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J. Büthe, An analytic method for boundingψ (x), Math. Comp. 87 No. 312 (2018), 1991-2009

work page 2018

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Chao and R

K.F. Chao and R. Plymen, A new bound for the smallestx withπ (x)− li (x)> 0, Int. J. Number Theory 6 No. 3 (2010), 681-690

work page 2010

[6] [6]

1 (2018), 227-251

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