{"id":"73835392-fbae-4773-8914-4330d2c4d62f","arxiv_id":"2501.04488","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"New error-term bounds improve Lehman's estimates for π(x) - li(x), removing the lower bound on the window width η and yielding sharper crossover regions near 10^316.","lead":"This paper sharpens the error bounds in Lehman's 1966 method for locating where the prime counting function π(x) exceeds the logarithmic integral li(x), and removes a constraint on the search window size. The improved estimates produce narrower certified crossover intervals near 10^316 and a longer run of consecutive integers where the difference remains positive.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of the new R6 error term in Theorem 3.1 is algebraically inconsistent: substituting (5.18)-(5.19) into (5.17) and summing over |γ|>A yields an extra factor of α relative to the claimed R6, so the written proof does not establish the theorem.","rationale":"The reader's weakest_assumption focused on the RH hypothesis, numerical zero accuracy, and interchange of summation and integration. However, the most load-bearing concern is an internal algebraic error in the derivation of R6, the central new contribution. The theorem's main claim (the improved R6 and the removal of the lower η bound) rests entirely on that error term. If the proof of R6 is invalid, the central claim is unsupported. The reader did not identify this issue, so I disagree with their choice of weakest assumption. A corrected derivation might salvage the theorem, but as written the paper does not establish its main result.","tokens_in":20890,"tokens_out":36781,"duration_ms":279041,"concrete_test":"Verify the algebra: write the right-hand side of (5.17), insert the bounds (5.18) and (5.19) with the stated constants, multiply by √(α/2π)∑_{|γ|>A} γ^{-2} (using Lemma 4.7a), and simplify. If the resulting expression is not bounded by the claimed R6 in (5.20), recompute with corrected constants and check which term is off. This requires only symbolic manipulation of the displayed inequalities, not new numerical computation.","verdict_should_be":"REJECT","load_bearing_attack":"Section 5.2 derives R6. From (5.17), the paper bounds the sum S by e A√α/(√e−1) e^{-α/4 η^2} (5.18) and the term T by A^2 e^{3/2} η/√α e^{-A^2/(2α)} (5.19). Substituting these into (5.17), then applying Lemma 4.7(a) with n=2 (∑_{|γ|>A} γ^{-2} ≤ 2A^{-1} log A) and the √(α/2π) prefactor, the first exponential term acquires a factor proportional to α^{3/2}/A log A, not the 8.283/A log A claimed in (5.20). The algebra between the penultimate display and (5.20) does not close: the factor A√α outside the bracket, combined with the √α/A inside the bracket, produces α, and the T term becomes Aη/α rather than Aη. No cancellation removes this α. Consequently the stated R6 is not a consequence of the written proof, and the claimed improvement over Lehman's S3 term is unverified.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21177,"tokens_out":15166,"duration_ms":133085,"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":[{"comment":"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":"Section 5.2, equations (5.17)–(5.20)"},{"comment":"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":"Section 5, equation (5.1)"},{"comment":"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.","section":"Section 7, Theorem 7.1"}],"minor_comments":[{"comment":"There is a typographical inconsistency: the condition is written 'w−η≥44.22' in one place and 'ω−η≥44.22' elsewhere.","section":"Theorem 3.1"},{"comment":"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":"Section 5.2, equation (5.20)"},{"comment":"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.","section":"Section 6.4"}],"recommendation":"major_revision","confidential_remarks":"The central algebraic error in Section 5.2 is concerning because it undermines the main theorem as stated. A corrected derivation may still yield a usable (and possibly better) R6 bound, so I do not recommend rejection; however, the authors must repair the proof and rerun the numerical tables before publication. The unproved Theorem 7.1 should also be either proved or removed from the claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nSoft 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.\n\nIn 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.","headline":"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.","tokens_in":70,"tokens_out":17106,"would_cite":false,"duration_ms":197582,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11N05","11Y11","11Y35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Prime counting function","Skewes numbers","Riemann zeros","Crossovers","Logarithmic integral","Explicit formula","Error bounds","π(x) − li(x) sign changes"],"falsifier":"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.","tokens_in":20691,"feed_emoji":"🔢","tokens_out":19748,"duration_ms":156225,"temperature":0.7,"pith_summary":"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 $10^{316}$.","feed_headline":"Certified prime-count windows shrink near 10^316","feed_subtitle":"Improved error terms shrink the certified interval near 10^316 by a third.","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original integrated formula, the Gaussian kernel, the $S_i$ error terms, and the sector/contour method that Theorem 3.1 refines.","marker":"[10]"},{"why":"Provides the 2010 sharp-region theorem and the prior error terms $S'_1,\\ldots,S_6$ that Theorem 3.1 improves, plus the numerical interval reworked in Section 6.3.","marker":"[13]"},{"why":"Gives the previous sharpest region, the kernel variant treated in Section 7, and the zero-sum Lemmas 2.9-2.10 used in the estimates.","marker":"[14]"},{"why":"Supplies the explicit prime-counting estimates used to derive (4.8) and the new $R_1$ term.","marker":"[6]"},{"why":"Provides the zero-counting estimate behind Lemma 4.6, used to bound sums over zeta zeros.","marker":"[2]"},{"why":"Introduces the numerical error analysis for approximate zeta zeros that yields the $\\Delta S_1$ and $\\Delta S_2$ corrections.","marker":"[17]"},{"why":"Gives the certified window near $10^{316}$ and the earlier crossover candidates that Section 8 compares against.","marker":"[3]"},{"why":"Establishes $\\pi(x)<\\mathrm{li}(x)$ for $x\\le10^{19}$, the starting range in the introduction.","marker":"[4]"}],"fun_headline_variants":["Lehman's prime-count error bound improved, eta condition dropped","Sharper sign-change bounds for pi(x)-li(x) without eta lower bound","Improved error terms shrink certified prime crossovers near 10^316","New bounds eliminate condition on eta, sharpen crossover regions","Refined Lehman estimates tighten prime-count sign-change intervals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Lehman's prime-count error bound improved, eta condition dropped","Sharper sign-change bounds for pi(x)-li(x) without eta lower bound","Improved error terms shrink certified prime crossovers near 10^316","New bounds eliminate condition on eta, sharpen crossover regions","Refined Lehman estimates tighten prime-count sign-change intervals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000696,"raw_usage":{"total_tokens":3223,"prompt_tokens":1096,"completion_tokens":2127,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":712,"completion_tokens_details":{"reasoning_tokens":2039}},"tokens_in":712,"tokens_out":2127,"duration_ms":15706,"temperature":1.0,"reasoning_tokens":2039,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:30:34.294082+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Lehman, On the difference π (x)− li (x), Acta Arithmetica XI (1966), 397-410","cited_arxiv_id":null,"evidence_quote":"Supplies the original integrated formula, the Gaussian kernel, the $S_i$ error terms, and the sector/contour method that Theorem 3.1 refines."},{"cited_title":"Saouter and P","cited_arxiv_id":null,"evidence_quote":"Provides the 2010 sharp-region theorem and the prior error terms $S'_1,\\ldots,S_6$ that Theorem 3.1 improves, plus the numerical interval reworked in Section 6.3."},{"cited_title":"Saouter, T","cited_arxiv_id":null,"evidence_quote":"Gives the previous sharpest region, the kernel variant treated in Section 7, and the zero-sum Lemmas 2.9-2.10 used in the estimates."},{"cited_title":"1 (2018), 227-251","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit prime-counting estimates used to derive (4.8) and the new $R_1$ term."},{"cited_title":"Backlund, Über die Nullstellen der Riemannschen Zetafunktion, Acta","cited_arxiv_id":null,"evidence_quote":"Provides the zero-counting estimate behind Lemma 4.6, used to bound sums over zeta zeros."},{"cited_title":"te Riele, On the sign of the differenceπ (x)−li (x), Math","cited_arxiv_id":null,"evidence_quote":"Introduces the numerical error analysis for approximate zeta zeros that yields the $\\Delta S_1$ and $\\Delta S_2$ corrections."},{"cited_title":"Bays and R.H","cited_arxiv_id":null,"evidence_quote":"Gives the certified window near $10^{316}$ and the earlier crossover candidates that Section 8 compares against."},{"cited_title":"Büthe, An analytic method for boundingψ (x), Math","cited_arxiv_id":null,"evidence_quote":"Establishes $\\pi(x)<\\mathrm{li}(x)$ for $x\\le10^{19}$, the starting range in the introduction."}],"review_version":1}