Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$

Jakob Ablinger

REVIEW 3 major objections 3 minor 16 references

Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$

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

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Pith's one-line read Two series for ζ(7) conjectured by Sun are now proven by an algorithmic reduction to cyclotomic polylogarithms.

desk verdict A credible computational proof of two Sun conjectures for zeta(7), with the decisive CAS reduction left as an unverified black box in the text and the new Section 3 identities unproved. read the letter →

arxiv 1908.06631 v1 pith:FOZM3I5P submitted 2019-08-19 math.CO cs.SC

Jakob Ablinger This is my paper · ORCID

classification math.COcs.SC MSC 05A1911M0611M3268W30

keywords infinitebinomialsumscentralcoefficientsharmonicnumberszeta(7)cyclotomicpolylogarithmsholonomicgeneratingfunctionscomputeralgebrasymbolicsummation

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 proves two infinite-sum identities conjectured by Zhi-Wei Sun: both sums, over $k$, of expressions built from harmonic numbers and the central binomial coefficient $\binom{2k}{k}$, are shown to equal the advertised rational linear combinations of $\zeta(7)$, $\zeta(2)\zeta(5)$, and $\zeta(3)\zeta(4)$. The proof works by viewing each series as the $x\to 1$ specialization of a generating function, deriving holonomic (polynomial-coefficient) recurrences and differential equations, solving those equations in terms of iterated integrals, and converting the result into cyclotomic harmonic polylogarithms at cyclotomy 3. Known structural relations among those polylogarithms then reduce the expression to the closed forms. The identities were conjectural; after this paper they are theorems, and the same pipeline produces further identities of the same shape.

What carries the argument

The load-bearing mechanism is the holonomic generating-function method combined with cyclotomic harmonic polylogarithms at cyclotomy 3. A holonomic sequence satisfies a linear recurrence with polynomial coefficients, and its generating function satisfies a linear differential equation; the package computes these from the summand, solves the differential equations in terms of iterated integrals (d'Alembertian solutions), and then rewrites those integrals using the substitution into the cyclotomic-polylogarithm alphabet with letters 0, 1, $\lambda=(3,0)$, and $\mu=(3,1)$. The final reduction is driven by shuffle, stuffle, multiple-argument, distribution, and duality relations among cyclotomic polylogarithms, which collapse the 243-term expression to the short closed form.

What would settle it

Evaluate both sides of (1) and (2) numerically with an independent arbitrary-precision implementation to, say, 100 decimal places; if any digit differs beyond the controlled rounding error, the central claim is false. Alternatively, run a certified summation algorithm on the two summands and check that the produced closed forms coincide with the paper's right-hand sides.

Watch

Extended reading notes

Core claim

The central claim is that the two Sun identities (1) and (2) are true exactly as stated. The author's derivation splits each summand into two parts, has the computer-algebra package compute their generating-function recurrences and differential equations, solves those equations as iterated integrals over $1/\tau$ and $\sqrt{\tau/(4-\tau)}$, and applies the substitution $\tau\to(\tau-1)^2/(1+\tau+\tau^2)$ to turn the result into 243 cyclotomic harmonic polylogarithms at cyclotomy 3. Applying shuffle, stuffle, multiple-argument, distribution, and duality relations reduces the first sum to $-\tfrac{459}{4}H_{0,0,1}H_{\lambda}^{4} - \tfrac{39}{2}H_{0,0,0,0,1}H_{\lambda}^{2} + \tfrac{45}{8}H_{0,0,0,0,0,0,1}$, which is exactly the right-hand side of (1); the second sum is reduced analogously to the right-hand side of (2). The paper also records several newly discovered identities of the same type in Section 3.

Load-bearing premise

The proof leans on the computer-algebra computations in Section 2 being correct: the recurrences, the differential-equation solutions, the 243-term cyclotomic-polylogarithm expression, and its relation-based reduction are each reported as command or solver output, without an independent derivation or a machine-checkable certificate, and the referenced notebook is not part of the submission.

Editorial extensions

If this is right

Sun's conjectures (1) and (2) are no longer open: both infinite binomial sums evaluate to the printed rational combinations of $\zeta(7)$, $\zeta(2)\zeta(5)$, and $\zeta(3)\zeta(4)$.
Any computation that uses those sums as numerical constants can now rely on exact closed forms rather than on extrapolated high-precision values.
The same generating-function-to-cyclotomic-polylogarithm pipeline yields additional identities of the same type, several of which are listed in Section 3.
The proof method is algorithmic: it applies in principle to any summand that is holonomic and whose generating-function solution lives in the iterated-integral and cyclotomic-polylogarithm class.

Reading between the lines

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

This suggests that infinitely many Sun-type binomial sums at odd zeta values should reduce to the same finite-dimensional space spanned by products $\zeta(7)$, $\zeta(2)\zeta(5)$, $\zeta(3)\zeta(4)$ and, for cyclotomy-3 variants, terms involving $\pi^7/\sqrt{3}$ and the constant $c=\sum_i 1/(3i+1)^4+1$; the Section 3 examples are consistent with this pattern.
A natural next step is to turn each solver output into a machine-checked certificate; without such certificates, the present proof's trust boundary sits inside the computer-algebra system.
The same substitution-and-relations strategy should be applicable to conjectural identities involving other cyclotomic alphabets, such as cyclotomy 5 or 7, where the number of letters and the dimension of the relation space grow but the pipeline does not change.

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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

The paper does what it says: it proves Sun's two conjectured binomial-sum evaluations for zeta(7), and it lists several new identities of the same type. The proof strategy is the author's own holonomic method: convert the summands to generating functions, derive differential equations, solve them in terms of iterated integrals, substitute to cyclotomic polylogarithms, and reduce using known relations. The recurrences and differential equations are displayed in full, and the final reduction from the cyclotomic-polylogarithm expression to (7) is plausible and consistent with the claimed zeta value. The two main identities are genuinely new proofs, not restatements of known results. The citation pattern is concentrated on the author's own prior work and Sun's conjecture list, which is normal for a niche computational method paper; I do not see a circularity problem, since the target identities come from outside the method.

The soft spot is exactly where the stress-test puts it. The 243-term cyclotomic-polylogarithm expression is shown only as six leading and six trailing terms with an ellipsis, and the reduction to (7) is asserted as the output of SpecialGLToH. No relation trace, no certificate, and no full expression appear in the text. The referenced notebook is external to the arXiv submission, so the reader cannot verify the hinge of the proof without downloading and trusting the package. This is a reproducibility gap, not evidence of a false result, but it is a gap at the exact point where a single wrong coefficient would change the conclusion. The Section 3 identities are presented as discoveries without proofs; they should be labeled as conjectures or supplemented with certificates. The convergence issue at x = 1 is also not discussed, though it is likely harmless because the sums converge and the method uses Abel summation; still, a sentence would settle it.

Overall, the core argument is sound in outline and the method is legitimate, but the paper as submitted is not fully checkable from its own text. The author should include the complete 243-term expression, the relation-based reduction, or a machine-checkable certificate, and should either prove the Section 3 identities or clearly mark them as unproved. This deserves a serious referee, because the main results resolve two published conjectures and the method is reproducible in principle with the named software.

Recommendation: send it to peer review, but with the expectation that the author provides enough computational detail to make the proof independently verifiable, not just asserted from a package run.

Referee Report

3 major / 3 minor

Summary. The paper claims proofs of two conjectured infinite binomial-sum identities due to Z-W Sun, stated as (1) and (2). The method follows the author's earlier framework: convert the summands into holonomic recurrences, pass to generating functions, solve the resulting differential equations, represent the solutions as iterated integrals, substitute to cyclotomic harmonic polylogarithms, and finally use algebraic relations among those polylogarithms to obtain the claimed zeta-value combinations. The paper also lists several additional identities that it says can be found by the same strategy. The proof chain is computer-assisted and relies on the author's HarmonicSums package.

Significance. If the proofs are correct and reproducible, the paper settles two open conjectures and demonstrates the utility of the author's holonomic-plus-cyclotomic-polylogarithm method for binomial sum evaluations. The additional identities in Section 3 are potentially useful. However, the value of the paper as a proof depends critically on the verifiability of the computer-algebra steps, and the manuscript as written does not make those steps checkable from its own text. The central mathematical claims are plausible and consistent with the cited conjectures, but the evidence supplied at the load-bearing reduction step is incomplete.

major comments (3)

[Section 2, Eq. (5)] The derivation of the generating-function representation (5) is asserted as the output of ComputeGeneratingFunction without showing the recurrence-solving process, initial-value comparisons, or the summation of the two holonomic solutions. Since (5) is the foundation for all subsequent substitutions and reductions, the reader cannot verify that the coefficients such as 4801781/73728 and 363/128 are correct. The paper should include a complete derivation or a machine-checkable certificate for this step.
[Section 2, SpecialGLToH application] The 243-term cyclotomic harmonic polylogarithm expression is only displayed as the first six and last six terms with an ellipsis. This expression is the hinge between the integral representation (5) and the final reduction (7). A single incorrect coefficient among the omitted 231 terms would change the final value and invalidate the proof. The full expression, or a verifiable certificate of its computation, must be supplied in the paper or in a stable ancillary file.
[Section 2, Eq. (7)] The decisive reduction of the 243-term expression to the five-term expression (7) is stated without any details: no relation trace, no basis transformation, and no certification. The sentence 'Applying these relations we find (7)' hides exactly the step on which the proof depends. The author should provide the full reduction, the list of relations used, or a separately checkable certificate (for example, a proof assistant output or an independent implementation of the same reduction).

minor comments (3)

[Section 3] The additional identities in Section 3 are presented without any indication of whether they are proven by the described method or merely discovered empirically. The paper should explicitly state their status; if they are proven, the necessary certificates or reproducible commands should be included, and if they are conjectural, they should be labeled as such.
[Notation in Eq. (7)] The notation H\lambda4 is not defined; it should be written as H_{\lambda,\lambda,\lambda,\lambda} or explicitly explained as a shorthand for four repeated indices. This would improve readability.
[Reproducibility details] The proof depends on an external notebook and a specific version of HarmonicSums. The paper should provide a persistent link, a checksum, or an appendix with the full session log so that referees and readers can reproduce the computations independently of the author's web page.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is an independent computer-algebra derivation against external conjectures; omitted verification certificates are a reproducibility concern, not a circular one.

full rationale

The central chain — summand → generating function → differential equations → d'Alembertian solution (5) → substitution → 243-term cyclotomic-HPL expression → relations to (7) → equality with the RHS of (1)/(2) — does not, on the paper's own equations, reduce to its inputs. The target identities are external conjectures from [16]; no parameter is fitted to the RHS, and the final equality is asserted as a checkable evaluation ('for which it is straightforward to verify that it is equal to the right hand side of (1)'), not as a definition of the RHS. The cited machinery (method of [2], relations in [6,8,10], and the HarmonicSums package) consists of general holonomic and cyclotomic-polylogarithm facts whose stated assumptions do not include the target sums. The one serious deficiency is that key reductions — particularly the compression of the truncated 243-term expression to (7) — are reported as CAS output without certificates or full listing. That is a reproducibility and trust concern, not a circularity: an omitted or erroneous coefficient would make the proof false, but it would not make the argument definitionally identical to its conclusion. Section 3's 'discovered' identities are additional outputs of the same independent computation, not predictions fitted to their own right-hand sides. Therefore no specific circular step can be quoted under the required standard.

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

The chain of identities relies on standard holonomic-function theory and on the correctness of the HarmonicSums algorithms. No free parameters are fitted; the user-defined constant c in Section 3 is a defined series, not an adjustable parameter.

assumptions (5)

standard math Holonomic sequences have holonomic generating functions and recurrences can be converted to differential equations.
Invoked in Section 2 where ComputeGeneratingFunction derives differential equations from the summand recurrences.
domain assumption The HarmonicSums differential equation solver correctly produces all d'Alembertian and Liouvillian solutions needed for the generating functions.
Section 2: 'These differential equations are solved using the differential equation solver implemented in HarmonicSums'; no independent derivation or certificate is included.
domain assumption The substitution tau -> (tau-1)^2/(1+tau+tau^2) transforms the iterated-integral representation into the stated cyclotomic-harmonic-polylogarithm representation.
Section 2 says 'we can use the substitution ... to compute a representation'; the 243-term output is shown only in excerpt.
standard math The shuffle, stuffle, multiple argument, distribution, and duality relations among cyclotomic polylogarithms are correctly applied by SpecialGLToH[7,3] to obtain the basis representation (7).
Section 2 cites [6,8,10] and [1, Section 4] for these relations; the reduction is asserted, not displayed.
domain assumption The limit x->1 of the generating function recovers the original infinite binomial sum.
Section 2 says 'after sending x->1 leads to (5)'; the convergence or analytic continuation argument is not given.

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Cite this review

Pith. "Pith review of Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$." pith.science (2026). https://pith.science/paper/FOZM3I5P

@misc{pith2026190806631,
  author       = {Pith},
  title        = {Pith review of: Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FOZM3I5P}},
  note         = {Machine review of arXiv:1908.06631}
}

read the original abstract

We give a proof of two identities involving binomial sums at infinity conjectured by Z-W Sun. In order to prove these identities, we use a recently presented method i.e. we view the series as specializations of generating series and derive integral representations. Using substitutions, we express these integral representations in terms of cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive the results. These methods are implemented in the computer algebra package HarmonicSums.

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Reference graph

Works this paper leans on

16 extracted references · 8 canonical work pages

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Pith tools

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

[1] [1]

Ablinger

J. Ablinger. Discovering and Proving Inﬁnite Binomial Su ms Identities. J. Exp. Math., 26, 2017. arXiv:1507.01703

arXiv 2017

[2] [2]

Ablinger

J. Ablinger. Discovering and Proving Inﬁnite Pochhammer Sum Identities. J. Exp. Math., 2019. arXiv:1902.11001

arXiv 2019

[3] [3]

Ablinger, Computing the Inverse Mellin Transform of Holonomic Sequences using Kovacic’s Algorithm

J. Ablinger, Computing the Inverse Mellin Transform of Holonomic Sequences using Kovacic’s Algorithm. in : PoS RADCOR2017, 069, 2017. arXiv:1801.01039

arXiv 2017

[4] [4]

Ablinger, Inverse Mellin Transform of Holonomic Seque nces

J. Ablinger, Inverse Mellin Transform of Holonomic Seque nces. PoS LL 2016, 067,

work page 2016

[5] [5]

Ablinger

J. Ablinger. The package HarmonicSums: Computer Algebra and Analytic as- pects of Nested Sums. in : Loops and Legs in Quantum Field Theo ry - LL 2014. arXiv:1407.6180

arXiv 2014

[6] [6]

Ablinger, J

J. Ablinger, J. Bl¨ umlein and C. Schneider. Generalized Harmonic, Cyclotomic, and Binomial Sums, their Polylogarithms and Special Numbers. J. Phys. Conf. Ser. , 523, 2014. arXiv:1310.5645

arXiv 2014

[7] [7]

Ablinger and J

J. Ablinger and J. Bl¨ umlein and C.G. Raab and C. Schneider . Iterated Binomial Sums and their Associated Iterated Integrals. J. Math. Phys. Comput , 55:1–57,

work page

[8] [8]

Ablinger, J

J. Ablinger, J. Bl¨ umlein and C. Schneider. Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials. J. Math. Phys. , 52, 2011. arXiv:1105.6063

arXiv 2011