{"id":"cdd774f9-a55f-454c-9c39-54af3c38bc1a","arxiv_id":"1908.01152","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A faster FFT-based algorithm computes the Kummer ratio for prime cyclotomic fields and establishes a new record maximum r(6766811)=1.709379041..., exceeding the old record at q=5231.","lead":"Using a fast Fourier transform trick, this paper computes the Kummer ratio for prime cyclotomic fields more quickly than before and reaches a new record value at the prime 6,766,811. A generalist should care because this gives numerical evidence about how class numbers behave for very large primes, a longstanding theme in number theory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Concern: the record r(6766811)=1.709379041... rests on FFT round-off without a certified error bound; the claimed independent cross-check is not shown, so the last digits are unverified.","rationale":"I independently checked the key identities: formula (11) follows from (8)-(10) with the usual first generalized Bernoulli number, and the decimation-in-frequency identity (12) is correct once the notation e(x)=exp(2*pi*i*x) is handled carefully. The small-q tables are consistent with the stated formulas, and no mathematical circularity or parameter fitting is present. The single load-bearing weakness is the numerical certification of the record value: the paper supplies no forward/backward error bound for the FFT, no side-by-side long-double versus quadruple values, and no reproducible table for the claimed Section 2 cross-check at q=6766811. This exactly matches the reader's weakest assumption. The concern does not overturn the verdict because the mathematical derivation appears sound and the record itself is separated from the old record by about 0.15, so it is robust to errors much larger than the displayed precision; however, the exact nine-decimal value and the formal status of the record remain conditional pending a concrete numerical verification.","tokens_in":11468,"tokens_out":23554,"duration_ms":232741,"concrete_test":"Recompute log r(6766811) using the digamma/cotangent method of Section 2 in quadruple precision and the Bernoulli method with an independent arbitrary-precision FFT (e.g., MPFR at 64 digits); publish both log r values to 15 decimal places along with the long-double value. If the maximum discrepancy exceeds 10^-9, the record digits r(6766811)=1.709379041... are not certified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of a new record maximum depends on the numerically evaluated sum in equation (11). Section 4.4 reports values computed in long double and quadruple precision, but the paper gives no round-off analysis for the FFT of length m=(q-1)/2 ~ 3.38e6. The inputs to the transform are of size O(1) and the outputs S_chi/q = B_{1,chi} are of size O(1/sqrt(q)), so cancellation amplifies relative errors by a large factor before logarithms are taken and summed over (q-1)/2 characters. A worst-case or even random-walk accumulation of per-operation rounding errors could reach the displayed digits. The manuscript asserts that the result was double-checked with the Section 2 method, but no such comparison is tabulated, and the two precisions are not shown side by side. Because the old record is 1.556562, a moderate error would not affect the record claim, but the exact value 1.709379041... is not certified. The paper is explicitly a preliminary report with non-archival code/data hosting, so an independent reader cannot verify the last digits from the printed material.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Kummer ratio r(q) = h_1(q)/G(q) for prime cyclotomic fields Q(ζ_q). It derives two exact expressions for log r(q): one involving digamma values (equation (6)) and one involving the first generalized Bernoulli numbers B_{1,χ} (equation (11)). The authors propose a decimation-in-frequency FFT strategy that reduces the sum over odd Dirichlet characters to a half-length transform, leading to an O(q log q) algorithm with O(q) logarithms and products. They report a new record value r(6766811) = 1.709379041..., exceeding the previous record r(5231) = 1.556562... of Shokrollahi, together with additional large values in Tables 3 and 4. The paper also contains scatter plots of r(q) for primes up to 2·10^6 and an application to Euler–Kronecker constants.","tokens_in":11683,"tokens_out":5336,"duration_ms":56089,"significance":"If the numerical record is correct, the paper provides a striking data point supporting the belief that r(q) is unbounded but only large on a rare sequence of primes. The algorithm is a genuine improvement over earlier direct summations, and the derivations of (6) and (11) are clean and rigorous, based on exact identities of Hasse and Shokrollahi. The paper contains no fitted parameters and the small-q computations are cross-validated with PARI/GP. The main weakness is the lack of certified numerical error bounds for the large FFT computations that underlie the record claim; the manuscript is explicitly a preliminary report and the code/data are hosted at a non-archival URL. These issues do not affect the mathematical derivation but do affect the reliability of the headline numerical result.","major_comments":[{"comment":"The record value r(6766811) = 1.709379041... is computed with floating-point FFTs, but the paper gives no round-off analysis for the O(q log q) operations at q = 6,766,811. The sums in (11) suffer from cancellation: the outputs S_χ/q = B_{1,χ} are of size O(q^{-1/2}) while the inputs are O(1), so relative errors can be amplified before the logarithm is taken. The manuscript reports long double and quadruple precision values but does not display them side by side, so the reader cannot see how many digits agree. To certify the displayed digits, the authors should provide a rigorous error bound (e.g., via interval arithmetic or a proven a posteriori bound) or at least a table showing the long double and quadruple values together with their difference.","section":"§4.4, Tables 3–4"},{"comment":"The sentence \"such a result was also double-checked using the method of section 2\" is not substantiated by any numerical comparison in the paper. Since the central claim is a new record maximum, the cross-check should be reported explicitly, for example by giving the value of r(6766811) obtained from the digamma method and the difference from the Bernoulli method. An unsupported assertion of a cross-check is not sufficient for a computational record.","section":"§3 and §4.4"},{"comment":"The manuscript identifies itself as a \"(preliminary) report\" and the computational data and programs are only available at a personal web page. For a formal journal publication, the numerical record must be reproducible from the article or from a stable, versioned repository (e.g., a DOI or an institutional archive). At minimum, the paper should state the software versions, compiler flags, and machine details needed to reproduce Tables 3 and 4, and ideally include the output data as supplemental material.","section":"Title footnote and §4.4"}],"minor_comments":[{"comment":"The title and running headers contain typographical errors: \"COMPUTA TION\" and \"RA TIO\" appear with stray spaces, and the author line includes a stray \"A\" after \"SEDUNOV A\". These should be corrected in the final version.","section":"Title and headers"},{"comment":"The Table 3 header includes \"n.a.\" (not available) but no entry in the table uses this value; either remove it or explain why it is listed.","section":"Table 3"},{"comment":"The footnote explaining the sign difference in the twiddle factor is confusing; the sentence \"we are now summing over the conjugate Dirichlet character χ instead over χ\" should be clarified, e.g., by writing \"instead of over χ\" and by explicitly indicating the conjugation in the formula.","section":"Footnote 2"},{"comment":"Reference [8] is an arXiv preprint from 2019; if this paper is published later, the final reference should include the journal publication data or a stable DOI, and access dates for the URLs should be provided.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The mathematical derivation is sound and the algorithm is a real contribution. The main obstacle is the numerical certification of the record value, which can be addressed by adding a rigorous error bound or a detailed cross-check table. The manuscript's self-description as a preliminary report makes the absence of such certification more significant than it would be in a final paper. I would be happy to reconsider after these additions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this is a competent computational note that delivers a new record for the Kummer ratio and an FFT-based algorithm that is genuinely faster than what was used before. The stress-test worry about certified error bounds is fair, but it affects only the last digits, not the record itself.\n\nWhat is new: formula (11) rewrites r(q) through first Bernoulli numbers, and the decimation-in-frequency FFT reduces the character sum to a half-length transform, giving O(q log q) products and O(q) logarithms. That is a real improvement over the O(q^2) naive summation used in earlier computations of this quantity. The headline result, r(6766811)=1.709379..., sits comfortably above Shokrollahi's old 1.556562. The paper also contributes tables up to q≈10^10, a scatter plot over two million primes, and a side computation of Euler-Kronecker constants. The derivations of (6) and (11) are straightforward and correct given the quoted theorems, and the citations look fair: the FFT ideas are credited to Ford-Luca-Moree and to Languasco's earlier note, and Shokrollahi gets the old record. No sign of citation padding.\n\nSoft spots are exactly where the stress-test points. The paper is explicitly a preliminary report, the code and data live on a personal webpage rather than an archive, and there is no floating-point error analysis for the FFT at half-length ~3.4 million. The claim that the result was double-checked by the digamma method is stated but not shown; Table 3 does not display long double and quadruple results side by side. So the last digits of the record value are not independently verifiable from the printed material. That is a real weakness, but it is a weakness in presentation, not a load-bearing flaw. The gap to the old record is about 0.15, which is orders of magnitude above plausible FFT round-off drift, and the fact that two precisions were used at all provides a sanity check.\n\nWho this is for: specialists in analytic number theory doing computations of cyclotomic class numbers or Euler-Kronecker constants. It deserves a serious referee. I would recommend sending it to review with requests for a cleaned numerical appendix: side-by-side results for both methods and both precisions, a worst-case or empirical error estimate for the FFT sums, and an archival hosting of code and data. Not a desk reject, not an accept as-is.\n\nBest,\n[Your name]","headline":"Solid computational note with a real new record and a clean FFT algorithm; the numerical certification is thinner than one would like, but the record claim itself is not in danger.","tokens_in":12242,"tokens_out":2175,"would_cite":false,"duration_ms":23793,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11-04","11Y60","11R18"],"pacs":[],"model":"deepseek-v4-flash","headline":"A half-length FFT computes the Kummer ratio near-linearly and sets a new record at q = 6,766,811.","keywords":["Kummer ratio","cyclotomic fields","class number","first factor","Bernoulli numbers","Fast Fourier Transform","decimation in frequency","Dirichlet characters"],"falsifier":"Run a certified interval-arithmetic version of formula (11) at $q=6766811$ and check whether the resulting interval contains $1.709379041$; if not, the record claim collapses.","tokens_in":11269,"feed_emoji":"🧮","tokens_out":7100,"duration_ms":65069,"temperature":0.7,"pith_summary":"This paper claims that the Kummer ratio $r(q)$ of the prime cyclotomic field $\\mathbb{Q}(\\zeta_q)$ can be computed in $O(q\\log q)$ multiplications and $O(q)$ logarithms by evaluating a closed-form expression that needs only a half-length discrete Fourier transform. Using this algorithm, the authors obtain a new record maximum $r(6766811)=1.709379041\\ldots$, surpassing the previous record $r(5231)=1.556562\\ldots$. The reduction rests on an identity expressing $\\log r(q)$ as a sum over odd Dirichlet characters of logarithms of the absolute values of a character sum, which a decimation-in-frequency FFT can evaluate at half the transform length. A sympathetic reader would care because this is the fastest known algorithm for a quantity that is believed to be unbounded yet rarely large, and the new record provides fresh computational evidence for that belief.","feed_headline":"Half-length FFT sets new record for class-number ratio","feed_subtitle":"The Kummer ratio can now be computed in near-linear time, and the largest known value jumps to 1.709 at q=6,766,811.","key_machinery":"The load-bearing identity is formula (11), which follows from a classical product formula for the first factor of the class number and from the first generalized Bernoulli number $B_{1,\\chi} = \\frac{1}{q}\\sum_{a=1}^{q-1} a\\chi(a)$; it expresses $\\log r(q)$ without any special function, using only the integer sequence $a=1,\\ldots,q-1$. The computational engine is the observation that the character sum is a prime-length DFT, combined with a decimation-in-frequency split that separates even and odd characters and reduces the required transform to half length; for $f(x)=x$ the split simplifies to $c_k = e(k/(q-1))(2a_k/q - 1)$.","core_discovery":"The paper establishes formula (11), $$\\log r(q) = \\frac{q-1}{2}\\left(\\log\\pi - \\frac{3}{2}\\log q\\right) + \\sum_{\\chi\\text{ odd}} \\log\\left|\\sum_{a=1}^{q-1} a\\chi(a)\\right|,$$ and shows that the inner sums over $a$ are discrete Fourier transforms of a sequence of length $q-1$. Because only odd characters are needed, a decimation-in-frequency strategy halves the transform length, so the total cost is $O(q\\log q)$ products and $O(q)$ logarithms. The authors implemented this with a standard FFT library and computed $r(q)$ for many primes, finding the new record value at $q=6766811$ and reporting several other large values near $1.7$.","pith_inferences":["The record value, if confirmed by a rigorously certified computation, would move the empirical maximum of $r(q)$ from $1.556$ to $1.709$ and sharpen the known lower bound for the limsup of $r(q)$ over primes; the paper itself does not make this formal claim.","The decimation-in-frequency reduction is a special case of a more general principle: any sum over odd characters of a DFT of a sequence can be computed at half length, which may speed up other class-number computations for characters of small order.","Because the input sequence $a=1,\\ldots,q-1$ is extremely smooth, a rigorous round-off error bound for the FFT at these sizes is likely attainable; if someone produces one, the numerical record becomes a theorem.","The scatter plot data, with roughly half of the $r(q)$ values above and below 1, hint at a near-symmetric distribution of $\\log r(q)$; a formal distributional theorem is not claimed here and would be a natural next step."],"forward_implications":["Primes far beyond the current range become searchable: with $q$ in the billions the computation takes only hours on a workstation, so larger record candidates are cheap to test.","The new maximum at $q=6766811$ and the repeated values near $1.7$ in Table 3 provide fresh computational evidence for the widely believed unboundedness of $r(q)$ on a thin set of primes.","The same half-length FFT decimation applies to the alternative digamma formula (6) and to the Euler–Kronecker constant difference $G_q - G_q^+$, so the algorithmic gain transfers to those quantities without new ideas.","The selection heuristic behind the large-prime search, that $r(q)$ tends to be large when $bq+1$ is prime for many small $b$, gave the right candidates and can guide future searches."],"supporting_citations":[{"why":"Supplies the product formula for $h_1(q)$ in terms of $B_{1,\\chi}$ and the previous record value $r(5231)$ that this computation surpasses.","marker":"[12]"},{"why":"Classical theorem expressing $h_1(q)$ as a product of $L(1,\\chi)$ over odd characters, the starting point of the derivation.","marker":"[6]"},{"why":"Provides the digamma-function formula for $L(1,\\chi)$, used in the second verification method.","marker":"[2]"},{"why":"Establishes that the character sums are prime-length discrete Fourier transforms, enabling the FFT approach.","marker":"[11]"},{"why":"Defines the first generalized Bernoulli number $B_{1,\\chi}$, foundational for formula (7).","marker":"[1]"},{"why":"Supplies the fast Fourier transform implementation used in the actual computations.","marker":"[3]"},{"why":"Earlier computations of the first factor of the class number, providing a benchmark for the data extension in Table 2.","marker":"[4]"}],"fun_headline_variants":["Near-linear Kummer ratio algorithm finds record at q=6.7M","FFT halves the work, sets class-number ratio record","Record Kummer ratio 1.709 via O(q log q) algorithm","Faster Kummer computation: new record 1.709 at q=6766811","Half-length FFT: Kummer ratio record jumps to 1.709"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The new record value depends on the untested accuracy of floating-point arithmetic in the FFT computation; the paper reports no error bound for round-off accumulation in the $O(q\\log q)$ operations.","fun_headline_variants_meta":{"raw":{"variants":["Near-linear Kummer ratio algorithm finds record at q=6.7M","FFT halves the work, sets class-number ratio record","Record Kummer ratio 1.709 via O(q log q) algorithm","Faster Kummer computation: new record 1.709 at q=6766811","Half-length FFT: Kummer ratio record jumps to 1.709"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000691,"raw_usage":{"total_tokens":3143,"prompt_tokens":974,"completion_tokens":2169,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":590,"completion_tokens_details":{"reasoning_tokens":2069}},"tokens_in":590,"tokens_out":2169,"duration_ms":14139,"temperature":1.0,"reasoning_tokens":2069,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:22:49.366793+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a certified interval-arithmetic version of formula (11) at $q=6766811$ and check whether the resulting interval contains $1.709379041$; if not, the record claim collapses.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the product formula for $h_1(q)$ in terms of $B_{1,\\chi}$ and the previous record value $r(5231)$ that this computation surpasses."},{"cited_title":"Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952, reprinted with an introduction by J","cited_arxiv_id":null,"evidence_quote":"Classical theorem expressing $h_1(q)$ as a product of $L(1,\\chi)$ over odd characters, the starting point of the derivation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the digamma-function formula for $L(1,\\chi)$, used in the second verification method."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that the character sums are prime-length discrete Fourier transforms, enabling the FFT approach."},{"cited_title":"Cohen, Number Theory","cited_arxiv_id":null,"evidence_quote":"Defines the first generalized Bernoulli number $B_{1,\\chi}$, foundational for formula (7)."},{"cited_title":"Frigo, S","cited_arxiv_id":null,"evidence_quote":"Supplies the fast Fourier transform implementation used in the actual computations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier computations of the first factor of the class number, providing a benchmark for the data extension in Table 2."}],"review_version":1}