{"id":"748d9e5e-32ba-41a2-9fe9-873923e8c052","arxiv_id":"2603.21555","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A Stieltjes-style limit formula for the Laurent coefficients Cn of the secondary zeta function about s=1 is derived, verified numerically, and accelerated via Brent's theorem.","lead":"The paper gives a limit formula for the regular-part coefficients Cn of the secondary zeta function Z(s) near its double pole at s=1, analogous to Stieltjes constants. It checks the formula numerically against known high-precision values and shows that Brent's error-bound technique improves the rate of convergence.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates RH as the sole foundational assumption and correctly classifies the paper as a solid but incremental contribution. Because that assumption is explicit, definitional, and standard for the secondary zeta function, it does not constitute a load-bearing flaw that would alter the conditional acceptance. The analytic steps are classical and the numerical evidence is consistent; therefore no adjustment of the reader’s verdict is warranted.","tokens_in":7775,"tokens_out":413,"duration_ms":4270,"concrete_test":"Independently recompute the n=0,1,2 instances of the plain limit (3) and of the BPT-corrected formula (44) with an independent list of the first 2·10^6 ordinates (e.g., Odlyzko or LMFDB) and compare the obtained digits against the 50-digit ADR table already published by the author; agreement to the predicted number of decimals confirms both the analytic identification and the error analysis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1) is a clean, expected generalization of Hassani’s n=0 limit to all Cn. The derivation proceeds by standard Stieltjes integration of N(T)=L(T)+Q(T) under RH, followed by the exp-log expansion of the remainder integral already present in Ivić and Bondarenko–Ivić–Saksman–Seip; the resulting integral representation of Cn matches the regular part of the known Laurent series. The RH hypothesis is stated from the first sentence and is definitional for Z(s) itself, not a hidden gap. Numerical checks with 2·10^6 zeros recover the ADR reference values to the precision predicted by the O(log^{n+1}T/T) error, and the BPT correction improves that error by an extra 1/T factor exactly as claimed. No internal inconsistency, circularity, or unsupported leap appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the secondary zeta function Z(s)=sum_gamma gamma^{-s} (sum over positive imaginary parts of non-trivial zeros, under RH). It recalls the known Laurent expansion of Z(s) about the double pole at s=1, with regular coefficients C_n, and proves a limit formula (Theorem 1) expressing each C_n as the T->infty limit of the partial sum sum_{gamma<T} log^n(gamma)/gamma minus an explicit main term built from the Riemann-von Mangoldt asymptotic. The proof proceeds by Stieltjes integration of N(T)=L(T)+Q(T), extraction of the main term A(T) by repeated integration by parts, and identification of the resulting constant with the regular part of the Laurent series via the exp-log expansion of the remainder integral already used by Ivic and by Bondarenko-Ivic-Saksman-Seip. Numerical checks with 2e6 zeros recover the first few C_n to the expected number of digits; an application of Brent-Platt-Trudgian (BPT) error bounds improves the remainder by an extra 1/T factor (Theorem 2) and is verified numerically.","tokens_in":7983,"tokens_out":875,"duration_ms":6752,"significance":"The main result is a clean, expected generalization of Hassani's n=0 limit (and of Brent's high-precision evaluation of C_0) to all regular Laurent coefficients C_n. The derivation is classical Stieltjes integration under RH and correctly matches the integral representation already present in the literature for the regular part of Z(s). The numerical verification against independent high-precision values obtained by the Arias de Reyna algorithm, together with the concrete BPT improvement, supplies a practical computational tool. The contribution is incremental rather than foundational, but it is self-contained, correctly executed, and of clear interest to specialists working on secondary zeta functions and sums over zeros.","major_comments":[],"minor_comments":[{"comment":"Throughout: the manuscript repeatedly writes 'Brent's (BPT) Theorem' and 'the (BPT) method'. BPT is the joint work of Brent-Platt-Trudgian; the attribution should be corrected for accuracy and consistency with the references.","section":null},{"comment":"Section 2, display (11)-(12): the lower-limit constant B_m is defined with a special case for m=0 that relies on the convention 0^0=1. A short clarifying sentence would remove any ambiguity for the reader.","section":null},{"comment":"Section 2, (16)-(17) and (21): several typographical slips appear (missing closing parentheses, 'Qt)' for Q(t), and an incomplete integral sign). These should be cleaned before publication.","section":null},{"comment":"Section 3, numerical checks: the text states that C_0 computed via ADR was 'offset by log^2(2pi)/(4pi)'. A one-line explanation of the origin of that offset (or a pointer to the earlier paper) would help the reader reconcile the two values.","section":null},{"comment":"Table 1 caption and surrounding text: the table is said to list coefficients 'to 50 digits' while the displayed entries for large n are given in scientific notation with fewer significant figures; a brief remark on the actual precision claimed for each entry would be useful.","section":null},{"comment":"References: several arXiv identifiers and journal citations are incomplete or slightly inconsistent in format; a uniform bibliographic style should be applied.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid, incremental contribution that correctly generalizes known n=0 results. It is appropriate for a number-theory journal that publishes work on zeta functions and sums over zeros; the main novelty is the uniform limit formula and the BPT improvement rather than a conceptual breakthrough. No circularity or hidden assumption beyond the openly stated RH appears. Minor copy-editing is all that is required."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that Theorem 1 is exactly what it claims: the natural Stieltjes-integration extension of Hassani’s n=0 formula to every regular Laurent coefficient Cn of the secondary zeta function about s=1. The derivation is the standard one already used by Ivić and by Bondarenko–Ivić–Saksman–Seip; the author simply inserts the log^m factor, identifies the constant term with the known series, and writes the limit explicitly. That is new in the literature, even if it is the expected next step rather than a conceptual leap.\n\nWhat the paper does well is keep the argument transparent and then check it. With 2 million zeros the raw sums recover the Arias-de-Reyna reference values to the number of digits predicted by the O(log^{n+1}T/T) error; adding the Brent–Platt–Trudgian correction improves the error by an extra 1/T factor and immediately yields 10–12 correct digits. The author also supplies a short table of high-precision Cn values obtained independently by the ADR algorithm, so the numerical side is reproducible and non-circular.\n\nSoft spots are minor and proportionate. Everything assumes RH from the first sentence, but that is definitional for Z(s) itself, not a hidden gap. The radius-of-convergence remark and a few lower-limit conventions are slightly pedantic but harmless. Self-citation of the author’s earlier ADR tables is legitimate because those tables are the only high-precision source available. No load-bearing flaw appears.\n\nThis is a short, solid note for the small community that actually computes constants attached to zeta zeros. It will not change anyone’s view of the secondary zeta function, but it gives a practical formula and a clean error analysis. I would send it to a referee; the math is standard, the checks are honest, and the result is useful inside its niche. Accept after light polishing.","headline":"Clean, expected generalization of Hassani’s n=0 limit to all Cn under RH, with a useful BPT acceleration and solid numerical checks; incremental but correctly done.","tokens_in":8560,"tokens_out":486,"would_cite":false,"duration_ms":4067,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11M06"],"pacs":[],"model":"grok-4.5","headline":"The regular Laurent coefficients of the secondary zeta function at its double pole s=1 are given by an explicit Stieltjes-style limit over the ordinates of the Riemann zeros.","keywords":["secondary zeta function","Laurent series","Stieltjes constants","Riemann zeros","Riemann hypothesis","zero sums","Brent theorem"],"falsifier":"Evaluate the partial-sum expression for C0 with the first 10^10 ordinates and compare the result, plus the explicit Brent error bound, against the independently known 19-digit value of C0; a discrepancy larger than the bound would refute the claimed limit.","tokens_in":8681,"feed_emoji":"🔢","tokens_out":889,"duration_ms":22903,"temperature":0.7,"pith_summary":"The secondary zeta function is the Dirichlet series formed from the positive imaginary parts of the non-trivial zeros of the Riemann zeta function, assuming the Riemann hypothesis. It possesses a double pole at s=1 whose Laurent expansion contains an infinite sequence of regular coefficients Cn. This paper proves that each Cn equals a concrete limit: the difference between a partial sum of log^n(γ)/γ over zeros up to height T and an elementary main term coming from the Riemann–von Mangoldt formula, taken as T tends to infinity. The expression is the precise analogue of the classical formula for the Stieltjes constants of the ordinary zeta function. Direct numerical checks with millions of zeros, further sharpened by Brent’s error-reduction theorem, confirm the formula to many decimal places and yield high-precision values of the coefficients.","feed_headline":"Limit formula recovers secondary-zeta Laurent coefficients","feed_subtitle":"A Stieltjes-style sum over Riemann zeros gives every regular coefficient at the double pole s=1.","key_machinery":"Stieltjes integration of the weight log^m(t)/t against the zero-counting function N(T)=L(T)+Q(T), which isolates an elementary antiderivative A(T) whose subtraction leaves a remainder that converges to (-1)^m Cm.","core_discovery":"For every integer n greater than or equal to zero the regular coefficient Cn in the Laurent series of the secondary zeta function about s=1 is recovered by the limit formula Cn = lim (T→∞) (-1)^n {sum_{γ<T} log^n(γ)/γ - [1/(2π(n+1)(n+2))] log^{n+1}(T) log(T^{n+1}/(2π)^{n+2})}.","pith_inferences":["The identical limit construction can be repeated at the simple poles of Z(s) that sit at the negative odd integers, producing analogous regular coefficients there.","The rapid growth of |Cn| visible in the computed table is consistent with a radius of convergence exactly equal to 2 and suggests factorial-type asymptotics.","Because the formula needs the Riemann hypothesis only up to height T, systematic comparison of the limit against independently computed Cn offers a practical numerical probe of the hypothesis itself."],"forward_implications":["The formula supplies an independent computational path to the coefficients Cn that does not rely on the Arias-de-Reyna algorithm.","Inserting the Brent–Platt–Trudgian correction improves the truncation error from O(log^{m+1}T/T) to O(log^{m+1}T/T^2), recovering many extra correct digits from a fixed zero database.","The same analysis yields an explicit integral representation of every Cn in terms of the oscillatory remainder Q(t).","High-precision tables of Cn for arbitrary n become available once sufficiently many ordinates are known."],"fun_headline_variants":["Limit formula recovers secondary-zeta Laurent coefficients from zero sums","Stieltjes-style sum over zeros yields secondary zeta regular coefficients","New limit formula gives all Cn for secondary zeta at the double pole s=1","Riemann zeros sum recovers secondary-zeta Laurent series regular part","Zero-sum limit formula extracts secondary zeta expansion coefficients"],"cache_read_input_tokens":128,"weakest_assumption_plain":"Every non-trivial zero is assumed to lie exactly on the critical line, so that its imaginary part is a real positive number that can be summed directly.","fun_headline_variants_meta":{"raw":{"variants":["Limit formula recovers secondary-zeta Laurent coefficients from zero sums","Stieltjes-style sum over zeros yields secondary zeta regular coefficients","New limit formula gives all Cn for secondary zeta at the double pole s=1","Riemann zeros sum recovers secondary-zeta Laurent series regular part","Zero-sum limit formula extracts secondary zeta expansion coefficients"]},"model":"grok-4.5","effort":"low","cost_usd":0.004772,"raw_usage":{"total_tokens":1270,"prompt_tokens":664,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":47720000,"prompt_tokens_details":{"text_tokens":664,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":536,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":664,"tokens_out":70,"duration_ms":4663,"temperature":1.0,"reasoning_tokens":536,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T20:16:23.881063+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Evaluate the partial-sum expression for C0 with the first 10^10 ordinates and compare the result, plus the explicit Brent error bound, against the independently known 19-digit value of C0; a discrepancy larger than the bound would refute the claimed limit.","supporting_citations":[],"review_version":1}