{"id":"9ca97cd6-5770-4996-a466-3fa6a44faf78","arxiv_id":"2608.01498","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"New explicit exponential- and log-type bounds for Mertens sums over primes, powered by bounds for weighted zero sums of the Riemann zeta function, improving on Vanlalngaia (2017) and correcting Dusart (2018).","lead":"This paper derives sharper rigorous error bounds for three classical prime-weighted sums, the sum of reciprocals of primes, the sum of (log p)/p, and the sum of Lambda(n)/n, plus the associated products over primes. The new bounds improve on Vanlalngaia (2017) and claim to repair a gap left by Dusart (2018), with extensive tables of constants.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorems 3(i) and 4(i) claim exponential decay constant 0.8746, but the proofs and eq. (152) only support C=0.84768; the stronger constant is never derived, so two headline theorems are unproved as stated.","rationale":"The reader's stated weakest assumption is the external Kadiri-Lumley-Ng zero-density estimate. That concern is legitimate but is a reliance on a published, cited result, and the manuscript does not exhibit an internal contradiction there. The more directly load-bearing issue is internal: two of the three headline theorems are printed with an exponential constant (0.8746) that the proof does not produce, and the conversion from the proven C=0.84768 form is impossible with a finite constant. This affects the central claim as stated, though the main mathematical contribution survives if the constant is corrected to 0.84768: the improvement over Vanlalngaia's 0.4183 is still by a factor of about 2.03, and the range extension to x >= 2 is unaffected. Therefore the reader's CONDITIONAL verdict is appropriate; the 0.8746 inconsistency should be fixed or explained, but it does not by itself invalidate the paper's core method. This is why I mark agreement as partial: the reader listed this issue as item (1) in the rationale but identified the external zero-density estimate as the weakest assumption, whereas I see the internal exponential-constant mismatch as the single most load-bearing concern for the stated theorems.","tokens_in":61705,"tokens_out":17766,"duration_ms":153715,"concrete_test":"Re-derive the final bound of Theorem 3(i) following eqs. (145)-(152), tracking every exponential constant. If the only constants that appear are C=0.84768 and C sqrt(2)=1.19878..., then replace every occurrence of 0.8746 in Theorems 3(i), 4(i), Tables 11-12, and Remark 1 with C=0.84768, and recompute the affected A_Upsilon/A_e_psi entries and the comparison factors with Vanlalngaia. If instead the intended constant is 0.8746, identify the zero-free region R=(2/0.8746)^2=5.228... used to define C in eq. (11), and verify that all Theorem 1 inputs and subsequent constants are recomputed with this R; otherwise the printed bounds are unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The stated decay constant 0.8746 in Theorems 3(i) and 4(i) (eqs. (17), (20)), Tables 11-12, and the Remark 1 comparison is not supported by the derivation. The proof chain is explicit: Proposition 28(iii) gives a zero-sum contribution D'(x0,sigma0)(log x)^B exp(-C sqrt(2) sqrt(log x)); combining with |theta(x)-x|/x <= A_theta(x0)(log x)^B exp(-C sqrt(log x)) in the proof of Theorem 3(i) via eq. (152) yields the final bound A_Upsilon(x0)(log x)^B exp(-C sqrt(log x)), with C=0.84768 from eq. (12). No step introduces 0.8746. This is not a harmless rounding issue: since exp((0.8746-0.84768) sqrt(log x)) is unbounded as x -> infinity, no finite constant A_Upsilon can convert the proven exp(-0.84768 sqrt(log x)) into exp(-0.8746 sqrt(log x)) for all x >= x0. Thus Theorems 3 and 4, and the comparison in Remark 1, overstate the exponential decay rate unless a different zero-free region with R = (2/0.8746)^2 ~ 5.228 is being used and propagated through Theorem 1. The same inconsistency appears in Remark 1 for lambda(x), which prints 0.8746 while Theorem 2(i) and eq. (14) use C=0.84768.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":61966,"tokens_out":6351,"duration_ms":59102,"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":[{"comment":"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.","section":"Theorems 3(i), 4(i) and §4, eqs. (17), (20), (152), (161); Tables 11–12; Remark 1"},{"comment":"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.","section":"Definition 11, eqs. (48)–(49); Proposition 26; Theorem 7"},{"comment":"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}].","section":"Proof of Theorem 3, eq. (151)"}],"minor_comments":[{"comment":"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}.","section":"Table 10 vs Theorem 2(i); Table 12 vs Theorem 4(i)"},{"comment":"The abstract contains the typo 'weighted sums of zeros of zeros of the zeta function'.","section":"Abstract and §1"},{"comment":"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.","section":"Table 1"},{"comment":"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.","section":"Remark 1, eq. (23)"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the same author group's prior estimates ([4], [12], [13], [17]) for the zero-density input and the ψ/θ bounds. This is not improper, but the dependence is strong and not independently verified, and the manuscript would benefit from an explicit statement of which inputs are from the same group and which are from independent groups. The main technical concerns are the unsupported 0.8746 decay constant and the dependence on the tabulated ZDB; both are fixable within the manuscript's scope, but the headline theorems must be restated with C=0.84768 unless a genuinely different zero-free region is introduced."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two headline exponential theorems in this paper state a decay constant 0.8746 that the proofs never establish. The derivations in Section 4 produce the bound with C=0.84768, the same constant as Theorem 1. No step gives 0.8746, and since exp((0.8746−0.84768)√log x) is unbounded, no finite A can bridge the gap. Tables 11–12 and Remark 1 repeat the unsupported constant. This is real, and it needs fixing before the paper can be used as stated.\n\nNow the credit: Theorem 2 for λ(x), the J_m zero-sum bounds (Theorem 7), the explicit formula (31), and the log-form tables are genuine work. The main chain—partial summation, explicit formula, zero-sum machinery—checks out structurally. The extension of Vanlalngaia's range from x≥e^4635 to all x≥2 is valuable, and the factor-2 gain in the decay constant survives with C=0.84768 for λ. The log-form bounds go to ℓ=5, which is more than the literature. The paper also flags correctly that Dusart's Mertens corollaries were damaged by the earlier ψ(x) mistake.\n\nSoft spots: the 0.8746 slip is the big one. It is probably a copy-paste from a different computation (maybe a different R), but it sits in the theorem statements, so readers cannot trust those two theorems as printed. Second, no code or data are shipped; the tables are extensive and the constants are explicit functions of inputs, so independent recomputation is possible but not immediate. Third, the 'recovered all of Dusart's results' claim is not reconciled row by row. Fourth, the zero-density estimate is cited, not re-derived; the whole exponential improvement flows through it. That is normal in this field, but the paper would be stronger if it stated the exact validity range and constants of the KLN estimate rather than deferring to [12, Table 7]. Minor typos (2.15 vs 2.18, a misplaced proof label in Theorem 3) lower clarity.\n\nWho is it for: anyone needing explicit bounds on Mertens sums/products, especially for applications in prime number theory, cryptography, or combinatorics. The log-form tables alone are a workhorse resource. It deserves serious refereeing; the 0.8746 issue is serious but repairable, and the rest of the edifice appears sound. My recommendation: send to a referee with instructions to check those two theorems and the tables, and to ask for a row-by-row reconciliation with Dusart. I would not cite it in its current form.","headline":"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.","tokens_in":62686,"tokens_out":5038,"would_cite":false,"duration_ms":42909,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N05","11M06","11M26"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Mertens sums","Mertens products","explicit bounds","Riemann zeta zeros","zero-density estimates","explicit formula","prime number theorem","Chebyshev functions"],"falsifier":"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.","tokens_in":61433,"feed_emoji":"🔢","tokens_out":12662,"duration_ms":104822,"temperature":0.7,"pith_summary":"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.","feed_headline":"Mertens decay constant doubles to 0.84768","feed_subtitle":"New explicit formulas over zeta zeros give best-known bounds on Mertens sums and products for every x ≥ 2.","key_machinery":"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","core_discovery":"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)","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the partial-summation identity (34) relating λ(x) to θ(x), the classical small-x Mertens bounds used to start the range, and the product arguments.","marker":"[32]"},{"why":"The 2017 predecessor whose exponential-form bounds for λ(x) and ψ̃(x) are improved by more than a factor of two and whose weighted-zero-sum method is extended.","marker":"[34]"},{"why":"The explicit zero-density estimate (ZDB) used to control zeros with real part above σ0.","marker":"[17]"},{"why":"Provides the telescoping zero-sum technique, the tabulated zero-density constants, and the bound A_ψ(2)=9.22022 used in Theorem 1.","marker":"[12]"},{"why":"Partial verification of the Riemann hypothesis up to H=3·10^12, removing low-lying zeros from the zero sums.","marker":"[28]"},{"why":"Supplies the zero-free region constant R=5.558691 entering every exponential decay rate.","marker":"[22]"},{"why":"Bounds for |ψ(x)-x|/x and the eta tables used to build the log-form constants.","marker":"[13]"},{"why":"Improved tables for θ(x) and eta constants used in Theorem 1 and in the numerical confirmation; also flags the 2018 error being corrected.","marker":"[4]"},{"why":"The lemma expressing averages of ψ(t)-t as zero sums, from which the Riemann–Guinand explicit formula is derived.","marker":"[30]"}],"fun_headline_variants":["New Mertens bounds sharpen exponential decay to C=0.84768","Explicit zeta-zero sums yield best-known Mertens inequalities","Mertens sums get sharper bounds via weighted zero density","Correcting Dusart: new bounds for Mertens sums and products","Sharp bounds for Mertens functions with improved decay constant"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["New Mertens bounds sharpen exponential decay to C=0.84768","Explicit zeta-zero sums yield best-known Mertens inequalities","Mertens sums get sharper bounds via weighted zero density","Correcting Dusart: new bounds for Mertens sums and products","Sharp bounds for Mertens functions with improved decay constant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000801,"raw_usage":{"total_tokens":3430,"prompt_tokens":890,"completion_tokens":2540,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":634,"completion_tokens_details":{"reasoning_tokens":2464}},"tokens_in":634,"tokens_out":2540,"duration_ms":18487,"temperature":1.0,"reasoning_tokens":2464,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T00:09:42.118879+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the zero-free region constant R=5.558691 entering every exponential decay rate."},{"cited_title":"Vanlalngaia","cited_arxiv_id":null,"evidence_quote":"The 2017 predecessor whose exponential-form bounds for λ(x) and ψ̃(x) are improved by more than a factor of two and whose weighted-zero-sum method is extended."},{"cited_title":"Kadiri, A","cited_arxiv_id":null,"evidence_quote":"The explicit zero-density estimate (ZDB) used to control zeros with real part above σ0."},{"cited_title":"Fiori, H","cited_arxiv_id":null,"evidence_quote":"Provides the telescoping zero-sum technique, the tabulated zero-density constants, and the bound A_ψ(2)=9.22022 used in Theorem 1."},{"cited_title":"Platt and T","cited_arxiv_id":null,"evidence_quote":"Partial verification of the Riemann hypothesis up to H=3·10^12, removing low-lying zeros from the zero sums."},{"cited_title":"Fiori, H","cited_arxiv_id":null,"evidence_quote":"Bounds for |ψ(x)-x|/x and the eta tables used to build the log-form constants."},{"cited_title":"Broadbent, H","cited_arxiv_id":null,"evidence_quote":"Improved tables for θ(x) and eta constants used in Theorem 1 and in the numerical confirmation; also flags the 2018 error being corrected."},{"cited_title":"Ramar´ e and Y","cited_arxiv_id":null,"evidence_quote":"The lemma expressing averages of ψ(t)-t as zero sums, from which the Riemann–Guinand explicit formula is derived."}],"review_version":1}