Infinite Summation Formulas Involving Riemann-Zeta function

Xiaoxia Wang; Xueying Yuan

REVIEW 3 major objections 3 minor 19 references

Infinite Summation Formulas Involving Riemann-Zeta function

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

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Pith's one-line read This paper establishes a series of new infinite summation formulas that express convergent series of generalized harmonic numbers weighted by central binomial coefficients as explicit combinations of Riemann zeta values and powers of ln 2.

desk verdict Routine coefficient-extraction paper with a load-bearing error in Section 3.2: Eqs (76)-(77) contradict the displayed expansion (75). read the letter →

arxiv 1908.09468 v1 pith:WWGJYDIR submitted 2019-08-26 math.CO

Xiaoxia Wang , Xueying Yuan This is my paper · ORCID

classification math.CO MSC 05A1033C20

keywords hypergeometricseriesinfinitesummationformulasRiemannzetafunctiongeneralizedharmonicnumberscentralbinomialcoefficientsGausstheoremWatsonBailey

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 derives exact infinite summation formulas that connect series of generalized harmonic numbers with central binomial coefficients to Riemann zeta values and powers of ln 2. Starting from Gauss, Watson, and Bailey hypergeometric summation theorems, the authors expand both sides as multivariate power series and compare coefficients term by term. If correct, each displayed identity is an exact evaluation of a convergent series, adding new members to the known family of zeta-related harmonic number identities.

What carries the argument

The load-bearing objects are the classical hypergeometric summation formulas of Gauss, Watson, and Bailey — exact evaluations of certain hypergeometric series at distinguished points — together with the gamma-function expansions (1)–(2) that convert product sides into exponential series in zeta values. The paper also uses the symmetric-function expansions (3)–(6) to rewrite finite products in terms of generalized harmonic numbers. Comparing the coefficient of each monomial a^i b^j c^k on both sides turns a hypergeometric identity into an infinite summation formula.

What would settle it

Numerically evaluate the left side of identity (15), the infinite sum of binom(2k,k)($O_k^{2}$ - $O_k^{{(2)}}$)/(k $2^{{2k}}$) over k, to high precision; if it does not equal 7ζ(3)/2, the coefficient-extraction claim fails.

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Extended reading notes

Core claim

The paper's central claim is that the displayed identities, such as (12)–(40), (46)–(59), (62)–(72), and (76)–(77), hold exactly. Each identity equates an infinite sum whose summand is a polynomial in the generalized harmonic numbers H_k and O_k (and their higher-order analogues) weighted by a central binomial coefficient or by 3^k, to an explicit combination of Riemann zeta values, powers of ln 2, and occasionally ζ(5). The identities are organized into three patterns: sums with binom(2k,k)/(k^i $2^{{2k}}$), sums with P_k/(k^i 2^k), and sums with 3^k/($k^{2}$ binom(2k,k)) multiplying a polynomial P_k.

Load-bearing premise

The argument assumes that the infinite sum over k and the formal power series in a,b,c can be expanded and compared coefficient-by-coefficient at (a,b,c)=(0,0,0) with no convergence or interchange obstacle.

Editorial extensions

If this is right

Each displayed identity gives a previously unknown closed-form evaluation of a convergent infinite series.
The coefficient-extraction method can be iterated to produce arbitrarily many new identities by taking higher-degree coefficient monomials.
The identities express certain odd zeta values, such as ζ(3) and ζ(5), as rational combinations of harmonic-number series, providing new representations of these constants.
The formulas also produce evaluations involving products of ζ(2) and powers of ln 2, enriching the family of harmonic-number identities.

Reading between the lines

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

If the coefficient-extraction step is made rigorous, the same procedure applied to higher-degree monomials in a,b,c should yield an infinite family of closed forms whose values are rational linear combinations of ζ(m) and powers of ln 2.
The appearance of ζ(5) alongside powers of ln 2 hints that the underlying generating function is a combination of polylogarithms at 1/2, which could connect these formulas to known Euler-sum tables.
One could test the method on a different classical summation theorem, such as Dixon's or Whipple's, and predict the exact form of the resulting harmonic-number identities before computation.

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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

Colleague,

The paper is a routine application of the hypergeometric coefficient-extraction method to generate infinite sums of harmonic numbers. The first two sections (Gauss and Watson) contain a large number of identities, some of which appear new, and spot-checks of a few early ones are consistent. But Section 3.2 has a load-bearing error: Eqs (76) and (77) do not follow from the displayed expansion (75). I checked the coefficient extraction for [b^2d] and [b^2d^2] by hand. The RHS in (75) is written with σ_k=ζ(k), correctly, since all gamma arguments are of the form Γ(1+·). That gives constants 26/3 ζ(3) and 40π^4/27. The paper instead prints 182/3 ζ(3) and 56π^4/3, which are exactly the values one gets by replacing σ_3 with τ_3=7ζ(3) and σ_2,σ_4 with τ_2,τ_4. So the authors appear to have extracted the coefficients against a different expansion than the one displayed. This is not a minor typo: it invalidates the two propositions in the third pattern, and it casts doubt on every identity in the paper that is asserted without showing the coefficient extraction.

What the paper does well: the method is standard and the first two sections produce many formulas in a systematic way. Some of these, e.g., (25)-(40) and (70)-(72), are not in the cited references. The paper is clearly written, and the reliance on known summation theorems keeps the derivations short.

Where it falls short: the derivation of every proposition is a one-line 'comparing coefficients' statement. No individual extraction is shown, so the reader cannot audit any single identity. That would be a presentation weakness; here it is worse, because it let the σ/τ confusion through. The novelty is also partial: some displayed identities are classical Euler sums (e.g., (62), (63)), and the whole approach is a continuation of the authors' earlier work [17,18]. There is no numerical or symbolic verification of the final formulas.

Who is this for: someone working in experimental mathematics or special-function software who wants a large table of harmonic-number sums might find parts of Section 2 useful, but they would need to verify each entry before using it. The paper as it stands is not reliable.

Recommendation: send it to peer review, but with a referee specifically asked to check a sample of the coefficient extractions. The two errors I found are concrete and easy to fix if the constants are corrected, but they are serious enough that the paper needs major revision before acceptance.

Referee Report

3 major / 3 minor

Summary. The manuscript claims to establish a large number of infinite summation identities involving generalized harmonic numbers and Riemann zeta values. The method is to substitute parameters into three classical hypergeometric summation theorems (Gauss, Watson, Bailey, and an additional summation theorem attributed to [16]), expand the resulting right-hand sides using the gamma-function expansions (1) and (2), expand the left-hand sides as multivariate Taylor series via the symmetric-function identities (3)–(6), and then compare coefficients term-by-term at the origin. The displayed identities are organized into three patterns: sums with central binomial coefficients and powers of 2, sums with denominator k^i 2^k, and sums with factor 3^k/(k^2 binom(2k,k)).

Significance. If all the displayed identities were correct, the paper would provide a useful catalogue of exact evaluations of harmonic-number series, and the use of classical summation theorems is a reasonable source of such formulas. The paper's breadth is its main strength, and some early identities pass numerical spot checks. However, the central claim that the listed formulas are exact evaluations fails as written: at least two displayed identities in Section 3.2 are quantitatively wrong. The paper also provides essentially no derivations, since every proposition is asserted after a one-sentence coefficient-comparison statement. Given the demonstrated extraction error, the lack of verifiable detail is a load-bearing weakness rather than a stylistic matter.

major comments (3)

[Sec. 3.2, Eqs. (74)–(77)] Propositions 3.9 and 3.10 are false as stated. The expansion (75) uses the coefficients σ_k from identity (1). For the coefficient of b²d, the right side of (75) gives −(26/27)σ_3 = −(26/27)ζ(3), while expansion of the left side of (74) gives [b²d] = −(1/9) Σ_{k≥1} 3^k(6O_k − 5H_{k−1})/(k² C_k), with C_k = binom(2k,k). Equating these gives Σ = 26/3 ζ(3), not 182/3 ζ(3). Similarly, for b²d², the right side of (75) has coefficient (120/81)σ_4 + (1/2)(8σ_2/9)² = 20π⁴/729, and the left side gives (1/54) Σ 3^k P_k/(k² C_k), where P_k is the polynomial displayed in (77); hence the sum should equal 40π⁴/27, not 56π⁴/3. The printed constants are exactly what one obtains by replacing σ_2, σ_3, σ_4 with τ_2, τ_3, τ_4. The derivation in this subsection therefore conflates the two gamma-function expansions (1) and (2), and the displayed identities must be corrected or removed.
[Throughout Secs. 2 and 3] No coefficient extraction is ever written out. Each proposition follows the sentence 'comparing the coefficients ... term-by-term' (after (11), (45), and (75)) without showing the actual multivariate expansion, the finite-product expansion, or the algebra for a single coefficient. This omission is not merely a presentation issue: the coefficient comparison in Section 3.2 is demonstrably inconsistent, so the reader cannot regard any unverified identity as trustworthy. The authors should provide complete derivations for all displayed identities, or supply a machine-checked verification of every identity, before the paper can be accepted.
[Sec. 3.1, Theorem 3.1 and Propositions 3.2–3.7] The term-by-term comparison at (a,b,c)=(0,0,0) requires justification that the hypergeometric series and the multivariate Taylor expansions commute and that the relevant sums converge uniformly in a neighbourhood of the parameter origin. This is particularly delicate for the Bailey theorem: after the substitution c → c+1, the expansion point c=0 lies on the boundary of the stated convergence condition R(c)>0 in (60). The manuscript contains no analytic argument for the interchange, and formal power-series equality alone does not establish convergence of the extracted infinite sums.

minor comments (3)

[Notation in Eqs. (14), (20), (40), (49), (65), (69)] Expressions such as '2ln2 2' and 'ln3 2' are ambiguous; they should be written as 2 ln²2, ln³2, and so on.
[Theorem 3.8 label] The theorem taken from reference [16] is called only 'summation theorem [16]' in the statement; naming it consistently would help readers.
[General presentation] Several displayed formulas obtained by linear combinations of earlier identities are unnumbered but are subsequently referred to by their constituent numbers; renumbering or labelling these would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the identities are obtained by standard coefficient extraction from external hypergeometric summation theorems.

full rationale

The derivation chain is a standard coefficient-extraction argument. The Gauss, Watson, Bailey, and nonterminating 3F2 summation theorems in (7), (41), (60), and (73) are external results quoted from [14] and [16]; the paper applies variable shifts and then compares multivariate power-series coefficients. The gamma-function expansions (1)-(2) come from [19], and the constants sigma_m and tau_m are explicitly identified with zeta(m) and (2^m-1)zeta(m); this identification is the mechanism that rewrites extracted coefficients as zeta values, not an assumption of the displayed identities themselves. No parameter is fitted and no displayed right-hand side is inserted as an ansatz. The only self-citation [17] appears in a survey sentence and is not load-bearing in any derivation step; Theorem 3.8 is credited to [16]. The formal term-by-term comparison lacks an analytic interchange justification, which is a rigor gap, and the apparent sigma/tau inconsistency affecting (76)-(77) would be an arithmetic error if confirmed, but neither makes the claimed derivation circular. Hence no circularity is present.

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

The paper is an application of external summation theorems and Gamma-function expansions. The identities inherit their zeta constants from expansions (1)-(2), where σ_m and τ_m are defined as ζ(m). No parameters are fitted, and no new entities are introduced. The main unproved step is the interchange of summation and coefficient comparison.

assumptions (6)

standard math Gauss, Watson and Bailey hypergeometric summation theorems hold for the shifted parameter values used.
Statements cited from Slater [14] as Theorems 2.1, 2.9, and 3.1; they are used as black-box inputs.
standard math The nonterminating 3F2 evaluation (73) from [16] is correct.
Cited as Theorem 3.8 and used without proof.
domain assumption Log-gamma expansions (1) and (2) with zeta-valued sequences σ_k and τ_k are valid for the arguments that occur.
Taken from Zheng [19]; these expansions determine the zeta constants that appear on the right-hand side of every identity.
domain assumption The two multivariate power series may be compared coefficient-by-coefficient term-by-term across the infinite sum in k.
The core methodological step after (11); convergence and interchange are not justified in the text.
standard math The symmetric-function expansions (3)-(6) correctly encode products of (1+x/k), (1-x/k)^{-1}, (1+y/(2k-1)), and their reciprocals.
Standard results from Macdonald [11], used to translate finite products into harmonic-number polynomials.
standard math Euler evaluations ζ(2)=π²/6 and ζ(4)=π⁴/90.
Used to express some right-hand side values in terms of π; not derived in this paper.

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Pith. "Pith review of Infinite Summation Formulas Involving Riemann-Zeta function." pith.science (2026). https://pith.science/paper/WWGJYDIR

@misc{pith2026190809468,
  author       = {Pith},
  title        = {Pith review of: Infinite Summation Formulas Involving Riemann-Zeta function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WWGJYDIR}},
  note         = {Machine review of arXiv:1908.09468}
}

read the original abstract

By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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Pith tools

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

[1] [16]

W ang, A nonterminating 7F6-series evaluation , Integral Transforms and Special Function

C. W ang, A nonterminating 7F6-series evaluation , Integral Transforms and Special Function. 29(9)(2018), pp. 719–724

work page 2018

[2] [1]

Ablinger, Discovering and proving inﬁnite binomial sums identities , Experimental Mathematics

J. Ablinger, Discovering and proving inﬁnite binomial sums identities , Experimental Mathematics. 26(1)(2017), pp. 62–71

work page 2017

[3] [2]

Andrews, R

G.E. Andrews, R. Askey and R. Roy, Special Functions, Cam bridge Univ. Press, Cambridge, 2000

work page 2000

[4] [3]

Berndt, Ramanujan’s Notebooks, Part I, Springer, N ew York, 1985

B.C. Berndt, Ramanujan’s Notebooks, Part I, Springer, N ew York, 1985

work page 1985

[5] [4]

Borwein and J.M

D. Borwein and J.M. Borwein, On an intriguing integral and some series related to ζ(4), Proc. Amer. Math. Soc. 123(4)(1995), pp. 1191–1198

work page 1995

[6] [5]

Chen and W

X. Chen and W. Chu, Dixon ’s3F2(1)-series and identities involving harmonic numbers and t he Riemann- Zeta function , Discrete Mathematics. 310(1)(2010), pp. 83–91

work page 2010

[7] [6]

Choi and H.M

J. Choi and H.M. Srivastava, Certain classes of inﬁnite series , Monatsh. Math. 127(1)(1999), pp. 15–25

work page 1999

[8] [7]

Choi and H.M

J. Choi and H.M. Srivastava, Explicit evaluation of Euler and related sums , Ramanujan J. 10(1)(2005), pp. 51–70

work page 2005