Computation of the Kummer ratio of the class number for prime cyclotomic fields

Alessandro Languasco; Alisa Sedunova; Pieter Moree; Sumaia Saad Eddin

REVIEW 3 major objections 4 minor 12 references

Computation of the Kummer ratio of the class number for prime cyclotomic fields

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A half-length FFT computes the Kummer ratio near-linearly and sets a new record at q = 6,766,811.

desk verdict 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. read the letter →

arxiv 1908.01152 v4 pith:MLZEJAAA submitted 2019-08-03 math.NT

Alessandro Languasco , Pieter Moree , Sumaia Saad Eddin , Alisa Sedunova This is my paper · ORCID

classification math.NT MSC 11-0411Y6011R18

keywords KummerratiocyclotomicfieldsclassnumberfirstfactorBernoullinumbersFastFourierTransformdecimationinfrequencyDirichletcharacters

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 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.

What carries the argument

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 $c_k = e(k/(q-1))(2a_k/q - 1)$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $1.7$.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

Dear colleague,

The 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.

What 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.

Soft 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.

Who 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.

Best, [Your name]

Referee Report

3 major / 4 minor

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.

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 (3)

[§4.4, Tables 3–4] 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.
[§3 and §4.4] 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.
[Title footnote and §4.4] 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.

minor comments (4)

[Title and headers] 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.
[Table 3] 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.
[Footnote 2] 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.
[References] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: r(q) is evaluated from exact class-number identities and an FFT, with no fitted parameters and no target value used as input.

full rationale

The paper's derivation chain is self-contained and non-circular. The Kummer ratio is defined by r(q)=h_1(q)/G(q) in equation (1); Hasse's theorem is then quoted to obtain r(q)=\prod_{\chi\,\mathrm{odd}} L(1,\chi) in equation (2), and the digamma and Bernoulli representations are quoted from [2] and [12], respectively. Equation (11), the main computational formula, is obtained by direct algebraic substitution: inserting B_{1,\chi}=(1/q)\sum a\chi(a) from equation (7) and Shokrollahi's product formula h_1(q)=2q\prod(-B_{1,\chi}/2) from equation (8) into the definition of r(q) yields equation (9), and taking logarithms yields equation (11). No parameter in this chain is fitted to the target values, and the reported record r(6766811)=1.709379... is an output of the fixed formula, not an input. The self-citations to [2] and [8] supply a standard digamma identity and an FFT decimation strategy, but both are parameter-free and are not used to enforce the claimed record; the decimation identity (12) is derived inline from primitive-root exponentiation and the discrete Fourier transform. The paper's stated limitations, namely that it is a preliminary report and that no rigorous round-off bound is given for the FFT evaluation in Section 4.4, are legitimate numerical-reliability concerns, but they are not circularity: an unquantified floating-point error would only affect certification of the last displayed digits, not the logical structure of the derivation. The asserted double-check via the Section 2 method is an additional reproducibility check, not a circular reuse of the same fitted data. Therefore no circular step is present, and the appropriate score is 0.

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

No free parameters or invented entities appear; r(q) is a fixed arithmetic function and all inputs are standard constants. The computation relies on several established theorems (Hasse, Shokrollahi, digamma representation) and on an unproved phase-cancellation step used to pass from complex to real logarithms.

assumptions (5)

domain assumption Hasse's class number formula h1(q)=G(q) ∏_{χ odd} L(1,χ)
Used in Section 2 to reduce r(q) to a product over odd Dirichlet characters; taken from Hasse [6] without proof.
domain assumption Shokrollahi's exact formula h1(q)=2q ∏_{χ odd}(-B_{1,χ}/2)
Used as Eq. (8) to replace L-values by first generalized Bernoulli numbers; cited from Shokrollahi [12].
domain assumption The identity L(1,χ)=-(1/q) ∑_{a=1}^{q-1} χ(a) ψ(a/q), eq. (3.1) of [2]
Basis of the digamma method in Section 2; cited from Ford-Luca-Moree [2].
standard math Every prime q has a primitive root g, so residues are enumerated by powers of g and character sums become DFTs
Elementary fact used in Section 4 to apply Rader's permutation and the FFT.
domain assumption The phase cancellation in Eqs. (5)-(6) and (10)-(11): the imaginary part of the sum over odd χ of log(...) equals -π(q-1)/2
Stated without proof in Sections 2-3; the real-valued formulas (6) and (11) depend on this cancellation.

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Cite this review

Pith. "Pith review of Computation of the Kummer ratio of the class number for prime cyclotomic fields." pith.science (2026). https://pith.science/paper/MLZEJAAA

@misc{pith2026190801152,
  author       = {Pith},
  title        = {Pith review of: Computation of the Kummer ratio of the class number for prime cyclotomic fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MLZEJAAA}},
  note         = {Machine review of arXiv:1908.01152}
}

abstract

Let $\zeta_q$ be a primitive $q^{\text{th}}$ root of unity with $q$ an arbitrary odd prime. The ratio of Kummer's first factor of the class number of the cyclotomic number field $\mathbb{Q}(\zeta_q)$ and its expected order of magnitude (a simple function of $q$) is called the Kummer ratio and denoted by $r(q)$. It is known that typically $r(q)$ is close to 1, but nevertheless it is believed that it is unbounded, but only large on a very thin sequence of primes $q$. We propose an algorithm to compute $r(q)$ requiring the evaluation of $O(q\log q)$ products and $O(q)$ logarithms. Using it we obtain a new record maximum for $r(q)$, namely $r(6766811) =1.709379\dotsc$ (the old record being $r(5231)=1.556562\dotsc$). The program used and the results described here, are collected at the following address \url{http://www.math.unipd.it/~languasc/rq-comput.html}. This is a (preliminary) report about the computational part of a joint project with Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Cohen, Number Theory

H. Cohen, Number Theory. Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics, vol. 240, Springer, 2007

work page 2007
[2]

K. Ford, F. Luca, P. Moree, Values of the Euler φ-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic ﬁelds, Math. Comp. 83 (2014), 1447–1476

work page 2014
[3]

Frigo, S

M. Frigo, S. G. Johnson, The Design and Implementation of FFTW3, Proceedings of the IEEE 93 (2), 216–231 (2005). The C library is available at http://www.fftw.org

work page 2005
[4]

G. Fung, A. Granville, H. C. Williams, Computation of the ﬁrst factor of the class number of cyclotomic ﬁelds, J. Number Theory 42 (1992), 297–312

work page 1992
[5]

Available from http://www.gnu.org/software/gsl/

Gnu Scientiﬁc Library, version 2.5, 2018. Available from http://www.gnu.org/software/gsl/

work page 2018
[6]

Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952, reprinted with an introduction by J

H. Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952, reprinted with an introduction by J. Martinet, Springer-Verlag, 1985

work page 1952
[7]

E. E. Kummer, Memoire sur la théorie des nombres complexes composes de racines de l’unité et des nombres entiers, J. Math. Pures Appl. (1851), 377–498, reprinted in his Collected papers, V ol. I, Springer-Verlag, 1975, pp. 363–484

work page 1975
[8]

Efficient computation of the Euler-Kronecker constants of prime cyclotomic fields

A. Languasco, A note on the computation of the Euler-Kronecker constants for prime cyclotomic ﬁelds, Arxiv, 2019, http://arxiv.org/abs/1903.05487

work page Pith review arXiv 2019

Show all 12 references

[9]

Moree, Irregular Behaviour of Class Numbers and Euler-Kronecker Constants of Cyclotomic Fields: The Log Log Log Devil at Play , Irregularities in the Distribution of Prime Numbers

P. Moree, Irregular Behaviour of Class Numbers and Euler-Kronecker Constants of Cyclotomic Fields: The Log Log Log Devil at Play , Irregularities in the Distribution of Prime Numbers. From the Era of Helmut Maier’s Matrix Method and Beyond (J. Pintz and M.Th. Rassias, eds.), S...

2018
[10]

Available from http://pari.math

The PARI Group, PARI/GP version 2.11.2, Bordeaux, 2019. Available from http://pari.math. u-bordeaux.fr/

2019
[11]

C. M. Rader, Discrete Fourier transforms when the number of data samples is prime, Proc. IEEE 56 (1968), 1107–1108

1968
[12]

M. A. Shokrollahi, Relative class number of imaginary abelian ﬁelds of prime conductor below 10000, Math. Comp. 68 (1999), 1717–1728. 2The minus sign here present in the twiddle factor e(−k/(q− 1)) comparing with the one in (12) depends on the fact that we are now summing over...

1999

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Cohen, Number Theory

H. Cohen, Number Theory. Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics, vol. 240, Springer, 2007

work page 2007

[2] [2]

K. Ford, F. Luca, P. Moree, Values of the Euler φ-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic ﬁelds, Math. Comp. 83 (2014), 1447–1476

work page 2014

[3] [3]

Frigo, S

M. Frigo, S. G. Johnson, The Design and Implementation of FFTW3, Proceedings of the IEEE 93 (2), 216–231 (2005). The C library is available at http://www.fftw.org

work page 2005

[4] [4]

G. Fung, A. Granville, H. C. Williams, Computation of the ﬁrst factor of the class number of cyclotomic ﬁelds, J. Number Theory 42 (1992), 297–312

work page 1992

[5] [5]

Available from http://www.gnu.org/software/gsl/

Gnu Scientiﬁc Library, version 2.5, 2018. Available from http://www.gnu.org/software/gsl/

work page 2018

[6] [6]

Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952, reprinted with an introduction by J

H. Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952, reprinted with an introduction by J. Martinet, Springer-Verlag, 1985

work page 1952

[7] [7]

E. E. Kummer, Memoire sur la théorie des nombres complexes composes de racines de l’unité et des nombres entiers, J. Math. Pures Appl. (1851), 377–498, reprinted in his Collected papers, V ol. I, Springer-Verlag, 1975, pp. 363–484

work page 1975

[8] [8]

Efficient computation of the Euler-Kronecker constants of prime cyclotomic fields

A. Languasco, A note on the computation of the Euler-Kronecker constants for prime cyclotomic ﬁelds, Arxiv, 2019, http://arxiv.org/abs/1903.05487

work page Pith review arXiv 2019