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REVIEW 3 major objections 3 minor 30 references

Fast Ramanujan-type Series for Logarithms. Part I

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A family of hypergeometric-type series computes log 2, log 3, and log 5 with lower binary-splitting cost than classical arctanh (Machin-type) formulas.

desk verdict Genuinely new fast log series with real software adoption, but the 'all but one proven' claim overreaches: only the d=2 cases are fully checkable in the manuscript. read the letter →

arxiv 2506.08245 v1 pith:H2USMTEC submitted 2025-06-09 math.NT

Jorge Zuniga This is my paper
classification math.NT MSC 11Y6033C20
keywords logarithmshypergeometricseriesRamanujan-typebinarysplittingWilf-ZeilbergermethodBetaintegralLLLalgorithmconvergenceacceleration
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

The paper introduces a family of hypergeometric-type series that compute logarithms of small integers, with the main examples being log 2, log 3, and log 5. The central claim is that these series, when evaluated with binary splitting, have lower computational cost per digit than classical Machin-type formulas, with the log 2 series at cost $C_s = 0.9679\ldots$ and similarly competitive costs for log 3 and log 5. The author reports proofs by Beta integral evaluations, hypergeometric closed forms, and rational Wilf–Zeilberger certificates, while explicitly flagging one of the log 2 series, Eq. (23), as conjectured. If the claim stands, these identities are directly usable in high-precision constants software, and the paper reports use in such software and calculation of over $10^{12}$ decimal places for some logarithms.

What carries the argument

The central object is a normalized hypergeometric-type convergent sum $\omega_N = \frac{1}{\beta}\sum_{n=1}^N R(n) H(n)$, where $R(n)$ is a rational function and $H(n) = \rho^n M(n)$ is a product of Pochhammer symbols; efficiency is measured by the binary-splitting cost $C_s = -4d/\log|\rho|$. The proofs flow through three tools: Beta integral representations that turn the sum into an integral of a rational function against $dx/\sqrt{1-x}$, closed-form evaluations of the hypergeometric sums, and rational Wilf–Zeilberger certificates produced by an $(s,t)$-WZ pair transformation.

What would settle it

Take Eq. (21) and evaluate it termwise to, say, $10^5$ terms with interval arithmetic, comparing the partial sum interval against known digits of $\log 2$ using the convergence bound from its rate $\rho = 1/(2^4 3^3 5^5)$; a mismatch at the expected precision would refute the identity. Independently, substituting the displayed WZ certificates into the required Wilf–Zeilberger recurrence and checking the identity by polynomial arithmetic would settle the $d=2$ proofs.

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Extended reading notes

Core claim

On its own terms, the paper claims that several newly found hypergeometric-type identities are the fastest known single-series algorithms for computing $\log 2$, $\log 3$, and $\log 5$ under the binary-splitting cost measure $C_s = -4d/\log|\rho|$. The fastest, Eq. (18), has $C_s = 0.9679\ldots$ and yields a linearly convergent series for $\log 2$; Eq. (19) and Eq. (20) give the fastest known single series for $\log 3$ and $\log 5$ with costs $1.4564\ldots$ and $1.2280\ldots$. Additional fourth- and sixth-degree identities, Eqs. (21), (22), (24), and (25), are presented as proven, with one series, Eq. (23), explicitly left as a conjecture. The proof methods are Beta integral evaluation, closed forms from hypergeometric sum evaluations, and rational certificates produced by a generalized Wilf–Zeilberger transformation.

Load-bearing premise

The load-bearing premise is that the computer-algebra certificates and Beta integral evaluations not shown in the paper are correct, since several proofs are stated to follow from them without details; if any certificate is wrong, the corresponding series is not proved.

Editorial extensions

If this is right

  • The identity Eq. (18) becomes a practical primary algorithm for high-precision $\log 2$ computation, replacing a Machin-type combination of arctanh series.
  • If the cost measure is right, computing $B$ bits of these logarithms takes time $O(M(B)\log B)$ with a small enough constant that existing digit records can be extended significantly.
  • The listed $d=4$ and $d=6$ series offer independent verification paths for log 2 and log 3, which is useful for cross-checking digits against a primary algorithm.
  • The parameterized identities Eqs. (64) and (65) show that the method extends to variable $p$, giving higher-order convergence as $p \to 1$.
  • The result provides a general template: integer-relation searches conjecture series of this shape, and Beta integral or WZ certificates prove them, so the same pipeline can be applied to other constants.

Reading between the lines

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

  • The paper leaves open how far the binary-splitting cost can be pushed for each logarithm; the same search could be run at higher deepness $d$ to look for even smaller $C_s$ values.
  • The alternating-series construction over complex quadratic fields produces only four rational cases up to $p=56$; searching other quadratic fields or parameter ranges could uncover additional rational fast series for larger primes.
  • Because the variable-$p$ series converge at rate $O((p-1)^{10})$ and $O((p-1)^{14})$, an obvious testable extension is to derive a whole ladder of $( u,m)$-balanced identities with convergence order growing linearly in $d$.
  • If the unproven Eq. (23) could be certified by the same WZ method, it would become another independent verification series for log 2; the paper suggests this is the 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.

Referee Report

3 major / 3 minor

Summary. The paper reports a family of linearly convergent hypergeometric-type series for log 2, log 3, and log 5, found by LLL integer-relation searches and WZ transformations. The main displayed identities are Eqs. (18)-(20) of depth d=2, Eqs. (21)-(24) of depth d=4, and Eq. (25) of depth d=6. The paper claims that, with the exception of Eq. (23), all displayed identities are proven by Beta-integral evaluations, hypergeometric closed forms, or WZ certificates, and that the selected series have lower binary-splitting costs than Machin-type formulas. It also reports incorporation of some series into y-cruncher and FLINT and states that over 10^12 decimal places were obtained for some logarithms.

Significance. If the identities and their proofs are correct, the d=2 series, especially Eq. (18), would be a practically valuable contribution to high-precision computation of log 2; Eqs. (56)-(58) provide explicit, checkable WZ certificates for the d=2 family, and the honest labelling of Eq. (23) as conjectured is a strength. The adoption of some identities in y-cruncher and FLINT and the reported digit records are useful external evidence, though not a substitute for proof. The principal weakness is that the proof claims for the d=4 and d=6 series are not substantiated within the manuscript as written.

major comments (3)
  1. [§5.2, §6.4] Eqs. (21), (22), and (24) are stated to be found and proven by named WZ transformations, but no WZ certificates R_{s,t}(n,k) are supplied for them; Section 6.4 explicitly says there is no room here to put their R_{s,t} certificates. A WZ proof consists of the certificate together with the rational-function identity it satisfies, so the reader cannot verify these three identities from the manuscript. Please include the certificates, or a supplementary file containing them, for every identity claimed to be WZ-proven.
  2. [§6.5, Example I] Section 6.5 leaves the d=4 Beta-integral proofs as an exercise to the interested reader, and Example I for Eq. (25) ends with the statement that Maple lines complete the proof, but those lines are not displayed. The proofs of Eqs. (21), (22), (24), and (25) are therefore not present in the manuscript. This contradicts the abstract's blanket statement that the found identities are proven and the Section 5 claim that all but Eq. (23) are proven; the proof status of half of the advertised fast series is unverifiable from the submitted text.
  3. [§7, Eqs. (64)-(65)] The variable-p series Eq. (64) and Eq. (65) are asserted to be proven by WZ and BI methods, but no certificates, partial-fraction data, or CAS output are provided, and Eq. (65) defines P(n,p) only through ellipses to be completed by symmetry. Since these formulas are part of the paper's claimed generalizations, either complete proofs or an explicit statement that they are asserted only conjecturally should be added.
minor comments (3)
  1. [§6.1-§6.5] After Eq. (35), after Eq. (42), after Eq. (45), and after Eq. (62), the text refers to Maple code or lines that are not included in the visible manuscript; even though the d=2 identities are already certified by Eqs. (56)-(58), the CAS-dependent passages should be made available for reproducibility.
  2. [Abstract, §5.1] The abstract says over 10e12 decimal places, while Section 5.1 mentions 3·10^12 decimal places for log 2; the notation 10e12 is ambiguous (usually meaning 10^13) and should be harmonized.
  3. [§5.2, Eq. (23)] Eq. (23) is clearly labelled as conjectured, which is a strength, but the abstract and Section 5 should avoid wording that could be read as claiming all displayed fast series are proven.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the fast logarithm series are discovered by lattice search but independently proven through Beta-integral, hypergeometric closed-form, and WZ-certificate arguments from external sources.

full rationale

The paper's central content is a set of hypergeometric identities for log p. The LLL/PSLQ search in Section 3 uses the target log p as a numerical probe to guess integer relations, but the paper does not present those guessed relations as predictions derived from the target. Instead, the displayed d=2 identities are consequences of the parameterized identity Eq. (45), which is derived from Hakimoglu's proven Eq. (43) and then evaluated by Beta Integral and hypergeometric closed-form methods, with WZ certificates Eqs. (56)-(58) supplied. Eqs. (21)-(22) and (24)-(25) are asserted to follow from WZ transformations of the source pair Eq. (17) and from the extended Beta Integral method; although the certificates and Maple lines are not displayed, the proofs are not obtained by fitting the target logarithm into the series coefficients. Eq. (23) is explicitly labeled 'just conjectured', so it is not a disguised derivation. The only self-citations, [28] and [29], concern the unproved status and the computational record of Eq. (23), and are not load-bearing for any mathematical derivation. Omitted WZ certificates and 'left as an exercise' BI details are proof-completeness concerns, not circularity under this rubric.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities; it constructs new series and a new WZ transformation. The only 'free parameter' is the unproven Eq. (23) coefficients. The main axioms are standard hypergeometric identities and the WZ convergence framework from the literature.

free parameters (1)
  • Integer coefficients of Eq. (23) = 686430, -742257, 223397, -13858
    Found via LLL search and admitted to be unproven; the coefficients are numerically detected rather than derived, so until a proof appears they behave as fitted parameters.
assumptions (6)
  • standard math Hypergeometric closed forms ΦA, ΦB, ΦC, ΦD from Sun-Zhou [7] are correct
    Used in Section 6.2 to prove d=2 identities by linear combinations of these forms.
  • standard math Hakimoglu's Eq. (2.9) [13] is correct
    Basis for the level-1 family Eq. (45) after substitution v = i u.
  • domain assumption The WZ method and 'flawless WZ pairs' [27] guarantee that WZ(s,t) transformations preserve the limit identities
    Section 4 relies on this to claim that new series in Section 5 are proven by WZ certificates.
  • ad hoc to paper Maple and other CAS symbolic integration outputs are reliable
    Several proofs in Section 6.5 are delegated to Maple code whose outputs are not shown in full.
  • standard math Dominated convergence permits interchanging sum and integral in the Beta integral derivations
    Used in Sections 6.1 and 6.5 to convert series to integrals.
  • domain assumption The binary splitting cost formula Cs = -4d/log|ρ| correctly ranks algorithm efficiency
    Used throughout to select 'fastest' series and to claim superiority over Machin-type formulas.

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Cite this review

Pith. "Pith review of Fast Ramanujan-type Series for Logarithms. Part I." pith.science (2026). https://pith.science/paper/H2USMTEC

@misc{pith2026250608245,
  author       = {Pith},
  title        = {Pith review of: Fast Ramanujan-type Series for Logarithms. Part I},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H2USMTEC}},
  note         = {Machine review of arXiv:2506.08245}
}
abstract

This report introduces new series and variations of some hypergeometric type identities for fast computing of logarithms $\log\,p$ for small positive integers $p$. These series were found using Wilf Zeilberger (WZ) method and/or integer detection algorithms (LLL) providing highly efficient linearly convergent rational approximants for these constants. Some of the new identities are of $_4F_3$ type, but higher ones are found as well and hypergeometric series for log p, with variable p, have been derived. Found identities are proven by I. classical Beta Integral methods, II. some hypergometric closed forms and III. rational certificates from the WZ method. Since they are very fast, these series are particularly suitable to be embodied in mathematical software being implemented in binary splitting form which produces very efficient algorithms. Over 10e12 decimal places have been obtained for some logarithms in reasonable time.

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Works this paper leans on

30 extracted references · 25 canonical work pages

  1. [1]

    Computing elementary functions using multi-prime argument reduction,

    Fredrik Johansson. Computing elementary functions using multi-prime argument reduction,. arXiv:2207.02501 [math.NA]. (2022). ( :1a:), (:1b:)

    arXiv 2022

  2. [2]

    A. K. Lenstra, H. W. Lenstra, L. Lov´ asz, Factoring polynomials with rational coefficients, Mathematische Annalen 261 (4) (1982) 515–534. doi:10.1007/BF01457454

    doi:10.1007/bf01457454 1982

  3. [3]

    Wilf, H. S. and Zeilberger, D. Rational Functions Certify Combinatorial Identities. J. Amer. Math. Soc. 3, 147-158, (1990) ( :3:)

    work page 1990

  4. [4]

    The Wilf–Zeilberger Method

    Koepf, W. The Wilf–Zeilberger Method. Ch. 6 in Hypergeometric Summation: An Algo- rithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 80-92, (1998)

    work page 1998

  5. [5]

    and Zeilberger, D

    Amdeberhan, T. and Zeilberger, D. Hypergeometric Series Acceleration via the WZ Method. Electronic J. Combinatorics 4, No. 2, R3, 1-3, (1997) ( :5:)

    work page 1997

  6. [6]

    Applied Mathematics E-Notes, 7 (2007), 237-246 ISSN 1607-2510 ( :6:)

    Gour´ evitch B., Guillera J., Construction Of Binomial Sums For π and Polylogarithmic Constants Inspired By BBP Formulas. Applied Mathematics E-Notes, 7 (2007), 237-246 ISSN 1607-2510 ( :6:)

    work page 2007

  7. [7]

    Zhou, arXiv:2401.14197 [math.CA] (2024), ( :7:)

    Zhi-Wei Sun, Y. Zhou, arXiv:2401.14197 [math.CA] (2024), ( :7:). Proof of Conjectures on Series with Summands Involving 2k k 8k/ 3k k 6k 3k

    arXiv 2024

  8. [8]

    A simple algorithm for the evaluation of the hypergeometric series using quasi-linear time and linear space

    S.V.Yakhontov. A simple algorithm for the evaluation of the hypergeometric series using quasi-linear time and linear space. arXiv:1106.2301 [cs.DS] (2011) ( :8:) F ast Ramanujan–type Series for Logarithms. Part I. 18

    work page Pith review arXiv 2011

Show all 30 references
  1. [9]

    Fast multiple-precision evaluation of series of rational numbers

    Haible B., Papanikolaou T. Fast multiple-precision evaluation of series of rational numbers. Proc. of the Third Intern. Symposium on Algorithmic Number Theory. June 21–25, (1998). pp. 338–350 ( :9:)

    1998

  2. [10]

    and Martin, C

    Cheng, H. and Martin, C. Evaluation of Modular Algorithms for High-precision Evaluation of Hypergeometric Constants. Computational Mathematics, Computational Geometry & Statistics (CMCGS 2012), (2012)( :10:)

    2012

  3. [11]

    Yee, Binary Splitting, Optimizing Recursions (2023) ( :11:)

    A. Yee, Binary Splitting, Optimizing Recursions (2023) ( :11:)

    2023

  4. [12]

    Cohen, Parametric Continued Fractions for π2, ζ(3) and other Constants, (2023) arXiv 2304.11727 [math.NT] ( :12:)

    H. Cohen, Parametric Continued Fractions for π2, ζ(3) and other Constants, (2023) arXiv 2304.11727 [math.NT] ( :12:)

    2023 arXiv

  5. [13]

    Cetin Hakimoglu, Accelerating the Hypergeometric Function with the Beta Integral to De- rive New Infinite Series for π and Values of the Gamma Function, arXiv:2402.08693v1 [math.CA] (2024) ( :13:)

    2024 arXiv

  6. [14]

    Helaman R. P. Ferguson, David H. Bailey, Steve Arno, Analysis of PSLQ, An Integer Re- lation Finding Algorithm, Mathematics Of Computation, Vol. 68, Number 22, Jan 1999, 351-369, S 0025-5718(99)00995-3 ( :14:)

    1999

  7. [15]

    A Polynomial Time, Numerically Stable Integer Relation Algorithm,

    H. R. P. Ferguson and D. H. Bailey, “A Polynomial Time, Numerically Stable Integer Relation Algorithm,” RNR Technical Report RNR-91-032, NASA Ames Research Center, MS T045-1, Moffett Field, CA 94035-1000 (1991)

    1991

  8. [16]

    and Schnorr, C

    R¨ ossner, C. and Schnorr, C. P. ”A Stable Integer Relation Algorithm.” Tech. Rep. TR-94-

  9. [17]

    FB Mathematik/Informatik, Universit¨ at Frankfurt, 1-11, (1994)

    1994

  10. [18]

    An html documentation of functions available under the GP/PARI CALCULATOR Version 2.17.0 (2024) ( :17a:) ( :17b:)

    2024

  11. [19]

    K. C. Au, Wilf–Zeilberger seeds and non-trivial hypergeometric identities, arXiv:2312.14051 [math.CO] (2024) ( :18:)

    2024 arXiv

  12. [20]

    Yee y-cruncher software (2025) ( :19:)

    A. Yee y-cruncher software (2025) ( :19:)

    2025

  13. [21]

    A. Yee. Records set by y-cruncher (2024) ( :20:)

    2024

  14. [22]

    FLINT : Fast Library for Number Theory (2024) ( :21a:) ( :21b:)

    Fredrik Johansson. FLINT : Fast Library for Number Theory (2024) ( :21a:) ( :21b:)

    2024

  15. [23]

    Maplesoft Maple Mathematics–based software (2024) ( :22:)

    2024

  16. [24]

    Hadrien Brochet, Bruno Salvy, Reduction-Based Creative Telescoping for Definite Summa- tion of D-finite Functions arXiv:2307.07216 [cs.SC] (2023) ( :23:)

    2023 arXiv

  17. [25]

    New series involving binomial coefficients

    Zhi-Wei Sun. New series involving binomial coefficients. arXiv:2307.03086 [math.NT] (:24:)

  18. [26]

    Euler’s Hypergeometric Transformations

    Weisstein, Eric W. Euler’s Hypergeometric Transformations. From MathWorld–A Wolfram Web Resource.(2024) ( :25a:), ( :25b:)

    2024

  19. [27]

    Scheibe, A

    P. Scheibe, A. Rich, N. Abbasi. RUBI, Rule-based integration: An extensive system of symbolic integration rules. Journal of Open Source Software vol 3-32, Dec. 2018 ( :26:)

    2018

  20. [28]

    The WZ method and flawless WZ pairs

    Jes´ us Guillera. The WZ method and flawless WZ pairs. arXiv:2503.00570v1 [math.NT], (2025) ( :27:)

    2025

  21. [29]

    How to prove these identities forlog(2) based on 3F2 integrals? MathOverflow (2024).( :28:)

    Jorge Zuniga. How to prove these identities forlog(2) based on 3F2 integrals? MathOverflow (2024).( :28:)

    2024

  22. [30]

    The Math of Performance: Stress Testing Hardware for Math-Changing Re- sults

    Jorge Zuniga. The Math of Performance: Stress Testing Hardware for Math-Changing Re- sults. StorageReview (2024). ( :29:)

    2024

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