REVIEW 3 major objections 3 minor 13 references
On an infinite number of nonlinear Euler sums
T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read This paper claims that eight families of quadratic Euler sums of odd order reduce completely to zeta values, polylogarithmic values at 1/2, and linear Euler sums, with the order-7 case left depending on only two unresolved constants.
desk verdict A real extension with many explicit sums, but the advertised order-7 reduction is incomplete and the order-5 results are asserted without derivations. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine is a collection of two-valued integer functions: identities that give closed forms for infinite sums indexed by a free integer parameter, such as sum_{k>=1} H_k^(2)/(k(k+i)) or sum_{i>=1} h_i/((i+k)^2), in terms of harmonic numbers of that parameter. Each such identity, when fed through partial-fraction decompositions and summation-order reversals, peels the quadratic sum down to zeta values and linear Euler sums. Many of these helper identities are imported from the authors' earlier preprints with phrases like 'analogously' or 'we found' rather than proved in the present text.
What would settle it
Numerically evaluate a representative asserted identity, say Eq. (56), at several integer values of i to high precision; or evaluate both sides of a stated order-7 result such as Eq. (141). A mismatch at the first unsupported identity would invalidate every order-7 result that uses it.
Extended reading notes
Core claim
The central discovery is a set of eight families of quadratic Euler sums of odd order — products such as H_k^(a) h_k^(b) divided by k^c, H_k^(a) H_k^(b) divided by (2k-1)^c, h_k^(a) h_k^(b) divided by k(2k-1), and similar — that can be evaluated in terms of zeta values, Li_n(1/2), and linear Euler sums of the form sum h_k^(n)/k^m. The order-3 case is expressed in zeta values and ln 2; the order-5 case in zeta values, Li_4(1/2), and Li_5(1/2); the order-7 case reduces every sum to zeta values and linear sums plus the two unresolved constants A = sum H_k^(2) h_k / $k^{4}$ and B = sum H_k h_k / $k^{5}$. The authors state that all members of each order-5 and order-7 family have been explicitly calculated in this sense.
Load-bearing premise
The entire chain depends on a large set of two-valued summation identities that are asserted without proof, mostly borrowed from earlier preprints; if any of them is wrong or merely approximate, the advertised reductions and all constants derived from them fail.
Editorial extensions
If this is right
- Order-3 and order-5 quadratic Euler sums of the eight families have fully explicit evaluations in zeta values and Li_n(1/2).
- For order-7, the eight families collapse to zeta values, linear odd-type sums, and the two unresolved sums A and B.
- Completing the reduction of A and B, likely with Li_7(1/2), would close the order-7 case and open order 9.
- The same two-valued-function scheme is announced as extendable to ternary Euler sums, with sample evaluations listed.
Reading between the lines
- Because the helper identities are asserted rather than derived, an independent computer-algebra verification of even a handful of them would either certify or refute the whole reduction chain.
- The two unresolved constants A and B may admit closed forms in terms of multiple zeta values or polylogarithms at 1/2; the paper's approximation to A is already accurate to about 3e-15.
- If the scheme generalizes as suggested, the same two-valued-function toolkit could be applied to even-order quadratic sums or to higher-degree products.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims a general method for evaluating eight families of quadratic Euler sums of odd order in terms of zeta values, polylogarithms Li_n(1/2), and linear Euler sums. The order-3 and order-5 families are presented as fully evaluated, while the order-7 families are reduced to zeta values, linear sums, and two unresolved sums A = ∑ H_k^(2)h_k/k^4 and B = ∑ H_k h_k/k^5; for A only a numerical approximation is given. The method rests on a collection of "two-valued" helper identities, many of which are imported from the authors' own preprints [10–12].
Significance. If the results were correct and complete, the paper would extend the known theory of nonlinear Euler sums to mixed products of even and odd hyperharmonic numbers and would provide a systematic reduction scheme for odd orders. The many explicit formulas in the paper could serve as useful benchmark values. However, because the central order-7 claim is explicitly incomplete and the auxiliary identities are not independently verified, the significance as a proof is substantially diminished.
major comments (3)
- [IV.A, V] The order-7 reduction is incomplete. The first-family formulas in Eqs. (42)–(49) express the sums in terms of zeta values, the linear sum ∑ h_k/k^3, and the two quadratic sums A = ∑ H_k^(2)h_k/k^4 and B = ∑ H_k h_k/k^5. Section V then states that "it remains to find a direct relation between" A and B in terms of zeta values and linear Euler sums, and supplies only the numerical approximation (164). Because A and B are products of harmonic and odd harmonic numbers, they are not "special types of linear Euler sums"; therefore the abstract's claim that the eight families are expressible by zeta values and linear Euler sums only is not established. The statements in §IV.B–G that all members have been "explicitly calculated" inherit this gap, since Eqs. (76), (77), (92), (100), and related formulas contain A or B.
- [V, Eq. (164)] The approximation (164), described as "correct within 2.9e-15", is not a proof and is not derived in the text. No explanation is given for the rational coefficients -1559/1943 and 1469/759, and a numerical agreement to fifteen digits, however suggestive, does not establish the required identity. Since this is the only evidence offered for the missing relation between A and B, the claim of a complete reduction to zeta values and linear Euler sums is unsupported.
- [III, IV] The derivations are largely omitted. In Section III no derivations are shown: each of Eqs. (15)–(40) is introduced with "we get" or "it follows" and no intermediate steps. In Section IV the auxiliary identities on which the reductions depend are either asserted ("we found") or referred to the authors' own preprints [10–12]; examples are Eqs. (56), (69), (96), (106), and (115). Since the entire chain of evaluations relies on these unproved two-valued help-function identities, the soundness of the advertised evaluations cannot be checked from the manuscript as it stands.
minor comments (3)
- [Abstract, Introduction] The text repeatedly uses "nominator" for "numerator" and "Salvi" for "Salvy" (reference [1]); the manuscript needs careful proofreading.
- [I, IV] The term "proper two-valued integer functions" is never defined in the present paper; the reader is left to infer its meaning from the references. A short formal definition would make the method self-contained.
- [Eq. (53)] The displayed identity for H^{(2)}_{k+i}/(k+i)^n appears garbled as printed; the right-hand side does not match the left-hand side as written. Please correct the formula or clarify the intended finite-difference relation.
Circularity Check
Order-seven 'explicit calculations' collapse into unresolved sums A and B, and the derivation leans on the authors' own preprints; the advertised reduction to zeta and linear sums is not closed.
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other
[Section IV.A, Eqs. (42)-(49); Section V (Summary), Eq. (164)]
"What remains is to find a direct relation between ∑∞ k=1 H(2)k hk/k4 and ∑∞ k=1 Hkhk/k5 in terms of zeta values and linear Euler sums. Up to now only the following approximation to ∑∞ k=1 H(2)k hk/k4 is known: ∑∞ k=1 H(2)k hk/k4 ≈ −1559/1943 ζ(2)ζ(5) + 1469/759 ζ(3)ζ(4). Remarkably, this results is correct within 2.9 ∗ 10−15."
The promised output for the order-seven families is an expression in zeta values and linear Euler sums only. Yet Eqs. (42)-(49) express the first-family members through the two quadratic sums A = ∑H^(2) h/k^4 and B = ∑H h/k^5; for example, Eq. (44) is a target quadratic sum expressed as zeta terms minus 2B minus A/2. The summary then admits that no exact relation is known and supplies only a numerically fitted approximation, Eq. (164). Sections IV.B-IV.G present 'all members ... calculated' with formulas still containing A and/or B, e.g. Eqs. (76), (77), (81), (83), (92), (100), (112), (124), (128). The claimed reduction to zeta values and linear Euler sums is therefore not delivered: the target nonlinear sums reappear as unresolved inputs, and the only closure offered is a fit.
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self citation load bearing
[Section IV.B and IV.C, Eqs. (75)-(83); also Eqs. (56), (69), (96), (106), (115)]
"All members of the second family can be calculated by use of lemma 2a from [11] and with the Euler sums belonging to the first family."
The cited lemmas come from the authors' own prior preprint [11], and the present text gives no proof or independent check of them; many further helper identities are introduced with 'analogously' or 'we found' (Eqs. (56), (69), (96), (106), (115)). These self-cited, unverified-in-this-text identities are load-bearing: they are what convert the first-family relations into the claimed order-seven evaluations. Since those evaluations still contain A and B, the self-citation chain does not close the advertised reduction; it shifts the missing step into the same authors' earlier work rather than providing an independent derivation.
full rationale
The order-three and order-five evaluations are explicit and, if the quoted helper identities are correct, have independent content; no definitional circularity is evident there. The serious problem is the order-seven part. In the paper's own final words, the reduction is incomplete: the two quadratic sums A = ∑H^(2) h/k^4 and B = ∑H h/k^5 remain unresolved, and Eq. (164) is only a numerical approximation, not an exact identity in zeta values and linear Euler sums. All later sections that claim to have 'calculated all members' inherit that missing closure, since their displayed formulas contain A and/or B. In addition, the derivation relies heavily on the authors' own preprints [10-12] for the two-valued helper identities, without proof or machine check in this text. This is partial circularity in the sense that the advertised outputs are re-expressed in terms of uncomputed quantities of the same nonlinear type, and the supporting identities are themselves borrowed from the same research program. No stronger definitional circularity is established: the paper does not simply rename a fitted parameter as a prediction, nor does it invoke a uniqueness theorem to force its ansatz. The result is therefore best scored as a 6: the order-seven 'calculations' reduce by the paper's own equations to the very type of nonlinear sums they were meant to eliminate.
Assumptions & free parameters
free parameters (1)
- Numerical approximation coefficients in Eq. (164) =
1559/1943 and 1469/759
assumptions (2)
- ad hoc to paper Two-valued help-function identities from [10-12] are correct, including Eq. (84) of [12], lemmas 2a/4a/4b of [11], and Eq. (160) of [11].
- domain assumption The unresolved sums A and B are eventually expressible in zeta values and Li7(1/2).
invented entities (1)
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Two-valued integer functions
Cite this review
Pith. "Pith review of On an infinite number of nonlinear Euler sums." pith.science (2026). https://pith.science/paper/SEHQUVVM
@misc{pith2026241200025,
author = {Pith},
title = {Pith review of: On an infinite number of nonlinear Euler sums},
year = {2026},
howpublished = {\url{https://pith.science/paper/SEHQUVVM}},
note = {Machine review of arXiv:2412.00025}
}
abstract
Linear harmonic number sums had been studied by a variety of authors during the last centuries, but only few results are known about nonlinear Euler sums of quadratic or even higher degree. The first systematic study on nonlinear Euler sums consisting of products of hyperharmonic sums had been published by Flajolet and Salvy in 1997 followed by similar studies presented during the last years by different authors. Although these studies had been restricted to sums where the nominator consists of a product of even or odd hyperharmonic sums, where the denominator is of the type $1/k^n$. We have generalized these results to nonlinear Euler sums with different denominators and nominators which consist in addition of mixed products between even and odd hyperharmonic numbers. In detail we present eight families of quadratic Euler sums which are expressible by zeta values and special types of linear Euler sums only where the order of the nonlinear Euler sums is always an odd number. The resulting eight different families of nonlinear Euler sums which we discovered consist of various products between even and odd hyperharmonic numbers, divided by three different types of denominators $1/k^n$, $1/((2k-1)^n)$ and $1/(k(2k-1))$. The calculational scheme is based on proper two-valued integer functions, which allow us to compute these sequences explicitly in terms of zeta values and pairs of odd-type linear harmonic numbers and even hyperharmonic numbers of second order.
Reference graph
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Reviewed August 12, 2026 · model on record in the stance chip above.
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