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Alternating multiple zeta values, and explicit formulas of some Euler-Apery-type series

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

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Pith's one-line read The paper establishes that every series in the families (4.1) and (4.2) has an explicit closed form in alternating multiple zeta values, with the single sums reducing to $\ln 2$ and zeta values.

desk verdict A serious, technically dense evaluation paper with a concrete false central formula in Theorem 2.4 (Eq 2.17); Section 3 is largely independent and looks sound, but the current version should not be used without correction. read the letter →

arxiv 1909.02943 v3 pith:GWXOQCCD submitted 2019-08-24 math.NT

Weiping Wang , Ce Xu This is my paper · ORCID
classification math.NT MSC 65B1011B6511M32
keywords alternatingmultiplezetavaluescentralbinomialcoefficientsgeneralizedharmonicnumbersiteratedintegralscontourintegrationRiemannlog-sineinversesums
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's project is to show that a natural family of infinite sums--terms with a central binomial coefficient divided by $4^n$, multiplied by generalized harmonic numbers--has closed forms rather than merely numerical values. The central claim is that every sum in the families (4.1) and (4.2) is an explicit finite rational combination of alternating multiple zeta values; within that, the single sums $S_p$ land in the smaller algebra $\mathbb{Q}[\ln 2,\zeta(2),\zeta(3),\ldots]$. If this is right, dozens of known and conjectural evaluations collapse into one bookkeeping rule, and the question of which constants appear is settled by the parity and sign structure encoded in the zeta arguments. The paper also conjectures that the wider family (1.1), with arbitrary products of harmonic numbers, admits the same reduction.

What carries the argument

The central object is the alternating multiple zeta value with barred and hatted entries: a multiple sum whose numerators carry powers of $(-1)^n$ and parity filters $1\pm(-1)^n$. Section 2 works through iterated integrals: the generating function $\sum_{n\ge0} \binom{2n}{n}4^{-n}t^n=(1-t)^{-1/2}$ and the harmonic-number analogue (2.19) convert each series into a simplex integral, while identity (2.5) expands parity-restricted multiple harmonic sums into $2^m$ sign choices, which become the barred arguments of the final zeta values. Section 3 uses contour integrals whose kernels are $\psi(-z)+\gamma$, $\pi^2\cot^2(\pi z)$, and related functions; the gamma-function expansion of Lemma 3.3 in polynomial coefficients turns residues into weighted products of zeta values, yielding the $\ln 2$-and-zeta reductions.

What would settle it

Check Eq. (2.17) at $m=1$: the series $S^\star_{1,1} = \sum_{n\ge1} H_n \binom{2n}{n}/(4^n n)$ has positive terms, while the displayed right-hand side is $-4\zeta(2)$; resolving this discrepancy against the value $2\zeta(2)$ quoted in Example 2.3 settles whether the hidden sign-string telescoping step is being carried out correctly.

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Extended reading notes

Core claim

On its own terms the discovery is a pair of reduction theorems. Theorem 2.2 states $S_{p+1} = -2\,\zeta(\bar 1,\{\hat 1\}_p)$. Theorem 2.3 states $$S_{m+1,p+1} = 4\,\zeta(\bar 1,\{\hat 1\}_p,\hat 2,\{\hat 1\}_{m-1}) - 2\,\zeta(m+1)\zeta(\bar 1,\{\hat 1\}_p),$$ where the barred and hatted arguments record sign and parity choices inside the alternating multiple zeta value. The contour-integral half of the paper proves the companion statements for the inverse binomial sums $\tilde S_q,\tilde S_{1,q},\tilde T_{1,q},\tilde U_{1,q}$, and in particular that each single sum $S_q$ is a polynomial in $\ln 2$ and Riemann zeta values. The final section packages these identities with an evaluation program, listing all series in (4.1)--(4.2) through weight 6.

Load-bearing premise

The load-bearing step is an unstated cancellation among sums over sign choices; if that cancellation is not valid, the Section 2 formulas do not follow.

Editorial extensions

If this is right

  • $S_{p+1}=-2\zeta(\bar 1,\{\hat 1\}_p)$ turns each single sum into one alternating MZV of weight $p+1$, so all low-weight values are immediately available.
  • The single sums $S_p$ are reduced constructively to a finite sum of products of $\ln 2$ and Riemann zeta values, not merely shown to lie in that algebra.
  • The inverse binomial sums $\tilde S_q,\tilde S_{1,q},\tilde T_{1,q},\tilde U_{1,q}$ inherit the same closed-form status, yielding identities such as $\tilde S_2=3\zeta(2)$.
  • All series in (4.1)--(4.2) through weight 6 are evaluated in the companion tables, making the reduction usable without re-deriving integrals.

Reading between the lines

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

  • The paper leaves implicit that its iterated-integral construction does not need the index string $\{1\}^m$; the same mechanism should extend to arbitrary harmonic index strings, giving the closing conjecture a proof path.
  • The contour-residue route computes residues through the polynomial coefficients of Lemma 3.2, so a purely algebraic reduction algorithm for inverse binomial sums at arbitrary weight should be possible.
  • A high-precision numerical check of one weight-7 series of type (1.1) with two distinct harmonic indices would test the closing conjecture before a proof is attempted.
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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 manuscript develops explicit evaluations for Euler-Apéry-type series involving central binomial coefficients and (generalized) harmonic numbers. Section 2 uses iterated integrals and alternating multiple zeta values (MZVs), while Section 3 uses contour integration with gamma, polygamma, and trigonometric kernels. The authors claim that the families in (4.1) and (4.2) reduce to (alternating) MZVs, with S_p reducible to ln(2) and zeta values, and they provide a Maple package and many worked examples. The paper's central message is seriously undermined by a false formula in Theorem 2.4, Eq. (2.17), which is inconsistent with the paper's own Corollary 2.5 and Example 2.3.

Significance. If the remaining formulas are correct, the paper would be a useful reference collection of explicit evaluations and would support a uniform MZV-reducibility statement for these Euler-Apéry-type series. The strengths are the large number of examples, the downloadable Maple package EASum, and the systematic organization of contour-integral and iterated-integral methods. No free parameters are fitted, and the identities are checkable numerically. The novelty is incremental relative to existing algorithmic packages such as HarmonicSums, and the main obstacle to acceptance is the correctness of Theorem 2.4.

major comments (3)
  1. [Section 2.4, Theorem 2.4, proof of Eq. (2.18)] Equation (2.17) is false as stated. For m=1, the left-hand side is S*_{1,1} = sum_{n>=1} H_n (2n choose n)/4^n/n, which is positive; by the paper's own Corollary 2.5 and Example 2.3 this sum equals 2 zeta(2), whereas the right-hand side is -4 zeta(2). For m=0, the right-hand side is the divergent quantity -2 zeta(1), while the series equals S_1 = 2 ln(2) by Theorem 2.2. The error is visible in the proof: Eq. (2.16) gives the integral as (-1)^m m! S*_{m,1}, so the prefactor in the displayed proof should be inverted, and the known evaluation integral_0^1 ln^m(t)/(1+t) dt = (-1)^m m! (1-2^{-m}) zeta(m+1) introduces an additional factor (1-2^{-m}). A corrected derivation yields S*_{m,1} = 2(2^m-1) zeta(m+1) for m>=1. Since S*_{m,1} is one of the target families in (1.3), this repair is necessary before the theorem can be used.
  2. [Section 2.4, Theorem 2.4, proof of Eq. (2.18)] The proof of Eq. (2.18) is only sketched as similar to the proof of Eq. (2.10), and it relies on the same sign-string telescoping whose neighboring derivation in Eq. (2.17) contains the inversion error described above. The current manuscript therefore does not provide enough detail to certify Eq. (2.18) as it stands. The authors should display the intermediate cancellations explicitly and verify the formula numerically for small cases, for example m=1, p=0 against Example 2.3, before the theorem is accepted.
  3. [Section 2.4, Theorem 2.4, m=0 case] The statement of Theorem 2.4 for all m,p>=0 is not compatible with Eq. (2.17): for m=0 the right-hand side is -2 zeta(1), which is divergent, whereas the left-hand series is S_1 = 2 ln(2). The range of parameters for Eq. (2.17) must be corrected to m>=1, or the m=0 case must be stated separately.
minor comments (3)
  1. [Section 2.4, Theorem 2.4, Eq. (2.17)] The theorem statement should also mention that zeta(m+1) is only meaningful in the usual sense when m>=1, and that the m=0 case requires a separate limiting or explicit evaluation.
  2. [Section 2.1, Eq. (2.5)] In Eq. (2.5), the range of the index j in the sum over sigma_j in {+1,-1} is not made explicit in the first displayed formula; the convention is clear from the following display but should be stated once.
  3. [Section 1, Introduction, unnumbered display after Eq. (1.3)] The representative formula for the series involving zeta*_n({1}^m) uses the symbols tilde and hat before their definitions are introduced in Section 2.1; the authors should define these notations earlier or move the display.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Euler-Apéry evaluations are derived from independent generating functions and elementary MZV identities; the self-cited supporting lemmas are not the target conclusions.

full rationale

The paper's central derivations (Theorems 2.2–2.9 and the contour-integral results of Section 3) start from independent inputs: the binomial generating functions (2.8) and (2.19) from [16,31], the standard expansion 1/sqrt(1-t), and the definitions of (alternating) multiple harmonic sums. No parameter is fitted to a set of evaluations, and no target series is inserted as a premise. The only self-citations in the proof chain are supporting identities: Eq. (2.3) is attributed jointly to [30] and [47], and Eq. (2.16) is cited to [47]. Both are elementary identities about Stirling numbers and log-integrals; they do not by themselves state the Euler-Apéry evaluations being proved, so invoking them does not make the conclusion equal to the input. The manuscript itself signals a compressed passage in Theorem 2.4 ('we present the crucial steps'), and the stated Eq. (2.17) appears to have a sign/constant defect: at m=1 its value -4ζ(2) contradicts the same paper's Corollary 2.5, which gives 2ζ(2). That is a correctness and verification risk, not a circularity, because the alleged reduction is not a renaming or a reuse of the claimed formula. No fitted input is relabeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and no formula is adopted by ansatz through a self-citation. Therefore no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper relies on prior generating functions and MZV identities, including two from the authors' own work, as computational lemmas. No empirical data or fitted constants enter; the only invented objects are the hat and tilde decorated mixed MZV symbols, which are not new entities but notation for parity-filtered sums.

assumptions (4)
  • domain assumption Generating function (2.8): sum (2n choose n)/4^n t^n/n = 2 ln(2/(1+sqrt(1-t))), cited from [16,31].
    Central starting point for the integral proofs in Section 2; assumed correct from prior literature.
  • domain assumption MHS and MZV identities (2.3) to (2.5), cited from [30,47], including the relation between zeta_n and signed sums with parity factors.
    The hat and tilde notation and the reduction of mixed MZVs to alternating MZVs rely on these identities.
  • standard math Residue theorem and kernel expansion Lemma 3.1 from Flajolet and Salvy; Legendre duplication formula and gamma asymptotics in Theorem 3.4.
    Contour integral evaluations in Section 3 assume standard complex analysis facts.
  • domain assumption Bell polynomial identities (2.14) to (2.16) for zeta*_n({1}^k), used in Theorems 2.4 and 2.9.
    Integral representation (2.16) is cited from the author's prior work [47]; a factor error there would change the results.

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Pith. "Pith review of Alternating multiple zeta values, and explicit formulas of some Euler-Apery-type series." pith.science (2026). https://pith.science/paper/GWXOQCCD

@misc{pith2026190902943,
  author       = {Pith},
  title        = {Pith review of: Alternating multiple zeta values, and explicit formulas of some Euler-Apery-type series},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GWXOQCCD}},
  note         = {Machine review of arXiv:1909.02943}
}
read the original abstract

In this paper, we study some Euler-Ap\'ery-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to ln(2), zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma functions, polygamma functions and trigonometric functions. The evaluations of a large number of special Euler-Ap\'ery-type series are presented as examples.

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