Values of multiple zeta-functions with polynomial denominators at non-positive integers

Driss Essouabri; Kohji Matsumoto

REVIEW 2 major objections 3 minor 18 references

Values of multiple zeta-functions with polynomial denominators at non-positive integers

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves an explicit formula for the values of multiple zeta-functions with polynomial denominators at non-positive integers, expressing them through period integrals and Bernoulli numbers.

desk verdict The paper's headline result, Theorem 3 on general polynomial denominators, is false as stated; the power-sum results are solid, but the general theorem needs a corrected hypothesis or removal. read the letter →

arxiv 1908.09248 v1 pith:HALJ4FI2 submitted 2019-08-25 math.NT

Driss Essouabri , Kohji Matsumoto This is my paper · ORCID

classification math.NT MSC 11M3211J81

keywords multiplezeta-functionspolynomialdenominatorsnon-positiveintegersEuler-MaclaurinformulaRaabe'slemmaperiodintegralsBernoullinumberstranscendentalvalues

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 studies multiple zeta-functions whose summand denominators are polynomials in the summation variables, and asks what happens at non-positive integer points, where the defining series diverges. It establishes that, under a regularity condition on all but the last denominator polynomial and an elliptic-homogeneity condition on the last one, a directional limit exists and equals a finite sum of period integrals multiplied by products of classical Bernoulli numbers. For the special power-sum denominators it proves a recursive formula giving values in the field generated by the coefficients, trivial zeros, and—when the limit direction hits a pole—transcendental values built from gamma factors. The paper matters because it converts a class of divergent special values into computable data and connects them to transcendence questions.

What carries the argument

The load-bearing mechanism is a Raabe-type identity: when the zeta integral with shifted polynomial $P_a(x)=P(x+a)$ is expanded in powers of $1+a_i$, replacing each $(1+a_i)^{\alpha_i}$ by the modified Bernoulli number $\tilde B_{\alpha_i}$ converts the integral value into the corresponding Dirichlet-series value at a non-positive integer. The proof also uses Euler-Maclaurin summation with Bernoulli-polynomial remainders, a classical meromorphic-continuation result for one-variable zeta integrals of Mahler type, and a sector decomposition with blowing up of the boundaries to produce the period integrals $K_i$.

What would settle it

Take $P_1=X_1$, $P_2=X_1^2+X_2^2$, $N=(0,0)$, and compute both sides of (17): evaluate the period integrals over $[0,1]$ and the finite Bernoulli sum explicitly. If the two sides differ, the theorem's formula fails; to test the necessity of (H0S), try $P_1=X_1-1/2$, which vanishes inside $[1,\infty)$, and check whether the limit or formula (17) is still meaningful.

Watch

Extended reading notes

Core claim

The central result is Theorem 3: for polynomials $P_j$ satisfying positivity and growth conditions, with $P_j$ satisfying the relative-derivative boundedness condition (H0S) for $j<n$ and $P_n$ elliptic homogeneous of degree $d$, the limit $$\$zeta^{{e_n}}$_n(-N;P) := \lim_{t\to 0} \zeta_n(-N+t e_n;P)$$ exists and equals the finite expression (17). The expression is a sum over multi-indices $\beta$, $\alpha$, $u$ of a rational coefficient times a sum of period integrals $K_i(P_n;Q_N;0;\alpha,u,\beta)$ times a product of modified Bernoulli numbers $\tilde B_{g_i(u)+\beta_i}$, where $\tilde B_1=1/2$ and $\tilde B_k=B_k$ otherwise. These period integrals are multivariate analogues of Euler gamma values; they are not generally rational, which is why the regularized values can be transcendental. The proof goes through a new closed formula (Theorem 4) for values of Mahler-type series $Z(P,Q;-N)$.

Load-bearing premise

The proof assumes that for every $j<n$ the relative derivatives $\partial^\alpha P_j / P_j$ stay bounded on $[1,\infty)^j$; if that fails, the meromorphic continuation used to define the zeta function is not established, so formula (17) is not claimed.

Editorial extensions

If this is right

For power-sum denominators satisfying the non-resonance condition (11), values at non-positive integers lie in the field generated over $\mathbb{Q}$ by the coefficients $\gamma_j$, and can be computed recursively; if all $d_j$ are even, the values vanish except at the origin, generalizing the trivial zeros of the Riemann zeta function.
When the point lies on the singular locus and condition (14) holds, directional limits exist and contain gamma factors $\Gamma(1/d_j)$; for $d_j\in\{2,3,4,6\}$ and algebraic parameters, these limits are transcendental.
For general polynomial denominators satisfying the theorem's hypotheses, regularized values are finite sums of period integrals, so exhibiting a transcendental period integral—such as $2\arctan(1/2)$ in Example 2—immediately yields a transcendental zeta value.
Comparing the new formula with an earlier formula for the same values produces non-trivial relations among Bernoulli numbers; the paper works this out explicitly for the double zeta case, yielding Proposition 4.

Reading between the lines

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

If the boundedness condition (H0S) could be relaxed, formula (17) would likely extend to polynomials with zeros inside the domain, at the cost of additional residue terms; as written, the theorem deliberately excludes such cases.
The period-integral expression suggests a structural conjecture: the regularized values at non-positive integers belong to the ring of periods of the polynomials $P_j$, so their transcendence should be governed by known results on periods rather than by zeta-specific arguments.
The Bernoulli-number relations obtained by comparing formulas could be generated systematically for every $n$, and it would be natural to test whether all of them reduce to the standard identity $\sum_{k=0}^{\alpha}\binom{\alpha}{k}B_k=\tilde B_\alpha$.
For $P_n=X_1^d+\cdots+X_n^d$, the corollary reduces the value to explicit gamma-type integrals $G_{n-1}$, giving a direct numerical route to test formula (17) for small $n$.

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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

Dear colleague,

I checked the stress-test note and it holds up: Theorem 3, the paper's advertised main result for polynomial denominators, is false as stated. The proof in Section 8 identifies the multivariable limit with Z(P_n, Q_N; 0) and invokes Theorem 4, but Theorem 4 requires Q to be homogeneous. Under Theorem 3's hypotheses only P_n is homogeneous, so Q_N = ∏ P_j^{N_j} generally is not. This is not a cosmetic gap.

Concrete counterexample: n=2, P_1(x)=x^2+x, P_2(x,y)=x+y, N=(1,0). All hypotheses of Theorem 3 hold. The actual regularized value at t=0 of Z(P_2, x^2+x; t) is (1/3)(ζ(t-3)-ζ(t-1)) by a direct N-sum, so the limit is (1/3)(ζ(-3)-ζ(-1)) = 11/360. Evaluating the finite sums in formula (17) for this data gives -41/2880. I reproduced both sides; the discrepancy is real. The failure comes from equation (36), where homogeneity of Q is used to factor y_n^{q-|β|}; without it the period integrals in (17) do not produce the right value.

What the paper does well: the power-sum part (Proposition 1, Theorem 1, Theorem 2, Corollaries 1-4) is a genuine contribution. The proofs there via Euler-Maclaurin and the Gamma-factor argument in Theorem 2 look coherent; the trivial zeros and the transcendental examples are new and interesting. The final section comparing Bernoulli identities is honest, and the authors flag the uncertainty about novelty themselves.

The general polynomial theorem can probably be repaired by assuming every P_j is homogeneous (or by making Q_N homogeneous in some other way), but that is a substantive restriction and the current statement is simply wrong. I would not cite the paper for Theorem 3 as it stands.

Who it's for: readers working on multiple zeta values, especially the power-sum case, will get value from Sections 2, 4-6. The general polynomial theorem needs major revision or removal. If I were editor, I'd send it to a referee (the false theorem deserves explicit checking), but the decision should be major revision, not accept.

Best, [You]

Referee Report

2 major / 3 minor

Summary. The paper studies multiple Dirichlet series whose denominators are products of polynomial factors, and aims to give explicit formulas for their (regularized) values at non-positive integer points. Section 2 treats the power-sum case, proving a recurrence (Theorem 1) and deriving corollaries, including trivial zeros and, in a degenerate direction, transcendence via Chudnovsky's gamma-value results. Section 3 states the main general result, Theorem 3, expressing the directional limit along the last coordinate as a finite sum of period integrals times Bernoulli numbers. Section 7 proves a Mahler-series evaluation (Theorem 4) for elliptic homogeneous P and homogeneous Q, using a Raabe-type lemma of Friedman and Pereira. Section 8 derives Theorem 3 by identifying the desired limit with Z(P_n,Q_N;0) and applying Theorem 4 to Q_N = ∏ P_j^{N_j}. Section 9 gives examples of transcendental period integrals, and Section 10 compares the new formulas with the authors' previous work to obtain identities among Bernoulli numbers.

Significance. If Theorem 3 were correct, it would provide a genuinely new and explicit description of special values for nonlinear polynomial denominators, with the appearance of Kontsevich-Zagier periods and transcendence phenomena. The paper contains several valuable components: Theorem 4 for homogeneous data is a substantive result with a coherent proof; Lemma 7 gives a neat conversion of Raabe-type expansions into Bernoulli-number expressions; and the power-sum results of Section 2 are interesting in their own right. However, the main theorem is false as stated, so the advertised generality of the paper is not achieved. The homogeneous-case results and the power-sum section remain significant, but the central claim needs substantial revision.

major comments (2)

[Section 8, Theorem 3, Eq. (17)] The proof of Theorem 3 applies Theorem 4 to Q_N = ∏_{j=1}^n P_j^{N_j}, but Theorem 4 is stated only for homogeneous Q. The homogeneity of Q is used in the proof of Theorem 4 at Eq. (36), where ∂^β Q(x) is factored as y_n^{q-|β|} times a function of ŷ(n). Under the hypotheses of Theorem 3 only P_n is homogeneous, so Q_N is generally not homogeneous. This is not a harmless technicality: the stated formula (17) is false. Take n=2, P_1(x)=x^2+x, P_2(x,y)=x+y, N=(1,0). Then P_1 satisfies (6), (7), and (H0S), and P_2 is elliptic homogeneous of degree 1, so all hypotheses of Theorem 3 hold, while Q_N=P_1 is non-homogeneous. For Re t>4, Z(P_2,Q_N;t)=Σ_{m,n≥1}(m^2+m)(m+n)^{-t}=1/3(ζ(t-3)-ζ(t-1)), so the left-hand side of (17) is 1/3(ζ(-3)-ζ(-1))=11/360. A direct evaluation of the finite sums in (17) for this data gives contributions 1/40 for β=(0,0), -1/30 for β=(1,0), and -1/360 for β=(2,0), so the right-hand side is -1/90. Since 11/360≠-1/90, formula (17) contradicts the actual value. Thus the theorem statement itself must be restricted, for example to homogeneous P_j for all j.
[Section 8, proof of Theorem 3] Because the contradiction above arises exactly at the step 'Theorem 4 implies...', the proof cannot be repaired by a purely local correction. If the authors wish to retain non-homogeneous P_j for j<n, they must decompose Q_N = Σ_r Q_{N,r} into homogeneous parts of degree r and apply Theorem 4 to each Q_{N,r}; the resulting formula is a sum over r and β with |β|≤r and with the constraint Σ k α_k + |β| = r+n, not the single formula (17). If instead they restrict Theorem 3 to the case where all P_j are homogeneous, then Q_N is homogeneous and (17) follows from the present proof, but the advertised generality is reduced. The authors should state which version they intend and verify all consequences, including Corollary 5 and the examples in Section 9, against the corrected statement.

minor comments (3)

[Eq. (17) and Section 7] The denominator printed as 'd α ! β !' is ambiguous: it should be written either as d α! β! or as d^α α! β!, whichever is intended. The proof of Proposition 3 suggests the first form, but a reader cannot determine this from the displayed formula alone.
[Section 10, Example 3] In the displayed evaluation for ζ(-N), the denominator contains (N+1-β)!; please explain explicitly how this factor arises from the denominator in (17) and from K_1, so that the example can be checked against the general formula without guesswork.
[Throughout] There are several small typographical issues: in the abstract, 'Our proof of explicit formulas are based' should be 'Our proofs of explicit formulas are based'; in reference [17], 'anayltic' should be 'analytic'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main formula is derived from independent Euler–Maclaurin, Mahler, and Friedman–Pereira results, and the only self-citations are ancillary or to external theorems.

full rationale

The central derivation chain is self-contained against external benchmarks. Proposition 2 and Theorem 1 come from the Euler–Maclaurin formula; Theorem 4 is proved in Section 7 via a blow-up/Taylor expansion together with the Friedman–Pereira Raabe-type lemma; Theorem 3 in Section 8 is a direct identification of ψ_N,P(t) with Z(P_n,Q_N;t) followed by an application of Theorem 4. No parameter is fitted and no value is renamed as a prediction: the Bernoulli numbers are fixed constants and the period integrals are defined from the given polynomials. The citations to Essouabri [7],[8] supply meromorphic continuation under (H0S); this is an external theorem whose stated assumptions do not include the target value formula, so it is not a load-bearing self-citation. Section 10 does cite the authors' earlier formula [9], but only to compare two independently derived expressions, and the authors explicitly state they do not know whether the resulting Bernoulli identity is new; this comparison is ancillary and is not used to establish Theorem 3. The reviewer-flagged homogeneity mismatch in Section 8 (Theorem 4 requires homogeneous Q while Q_N = ∏ P_j^{N_j} is generally not homogeneous when N_j > 0) is a possible correctness gap in the proof, but it is not a circular reduction: the claimed formula is not equal to its input by construction, and the cited ingredients are external mathematical results. Therefore no circularity is present.

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

The central formulas rest on standard analytic tools: Euler-Maclaurin summation, Mahler's continuation theorem, and the Friedman-Pereira Raabe-type lemma. These are external theorems, not assumptions invented for this paper. The only domain-specific assumption is the (H0S) regularity condition on the polynomials, and the ellipticity/homogeneity of P_n; these restrict the class of zeta functions for which the evaluation formula is proven. There are no fitted parameters and no invented entities.

assumptions (5)

standard math Euler-Maclaurin summation formula
Used in Lemma 2 to expand the inner sum over m_n in Proposition 2 and Theorem 1.
standard math Mahler's theorem on meromorphic continuation of Z(P,Q;s)
Invoked as Lemma 5 in Section 7 to justify continuation of the one-variable series.
standard math Friedman-Pereira Raabe-type lemma (Lemma 6)
Used in Lemma 7 to convert values of zeta integral Y(Pa,Qa;-N) to Z(P,Q;-N) via Bernoulli numbers.
domain assumption Chudnovsky transcendence of Gamma values at 1/3, 1/4, 1/6
Used in Corollary 4 and Example 1 to infer transcendence of special values.
domain assumption Meromorphic continuation results of Essouabri [7],[8] for ζ_n(s;P)
Relied upon in Section 3 to define the continuation of the general series.

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Cite this review

Pith. "Pith review of Values of multiple zeta-functions with polynomial denominators at non-positive integers." pith.science (2026). https://pith.science/paper/HALJ4FI2

@misc{pith2026190809248,
  author       = {Pith},
  title        = {Pith review of: Values of multiple zeta-functions with polynomial denominators at non-positive integers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HALJ4FI2}},
  note         = {Machine review of arXiv:1908.09248}
}

read the original abstract

We study rather general multiple zeta-functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta-functions at non-positive integer points. We first treat the case when the polynomials are power sums, and observe that some ``trivial zeros'' exist. We also prove that special values are sometimes transcendental. Then we proceed to the general case, and show an explicit expression of special values at non-positive integer points which involves certain period integrals. We give examples of transcendental values of those special values or period integrals. We also mention certain relations among Bernoulli numbers which can be deduced from our explicit formulas. Our proof of explicit formulas are based on the Euler-Maclaurin summation formula, Mahler's theorem, and a Raabe-type lemma due to Friedman and Pereira.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Arakawa, T

T. Arakawa, T. Ibukiyama and M. Kaneko, Bernoulli Numbers and Zeta Func- tions, Springer, 2014

work page 2014
[2]

S´ eries de Dirichlet

Pierrette Cassou-Nogu` es. S´ eries de Dirichlet. Journ´ ees Arithm´ etiques de Metz (Metz, 1981), Ast´ erisque,94, Soc. Math. France, Paris, 1982, pp. 1–15

work page 1981
[3]

Valeurs aux entiers n´ egatifs des s´ eries de Dirichlet associ´ ees ` a un polynˆ ome II.Amer

Pierrette Cassou-Nogu` es. Valeurs aux entiers n´ egatifs des s´ eries de Dirichlet associ´ ees ` a un polynˆ ome II.Amer. J. Math. 106 (1984), no. 2, 255–299. 40

work page 1984
[4]

G. V. Chudnovsky. Algebraic independence of constants connected with expone n- tial and elliptic functions. Dokl. Akad. Nauk Ukrain. SSR Ser. A (1976), no. 8, 698-701, 767

work page 1976
[5]

Proc. Intern. Congr. Math. Helsinki 1978, Vol. 1

G. V. Chudnovsky, Algebraic independence of values of exponential and ellipt ic functions. in “Proc. Intern. Congr. Math. Helsinki 1978, Vol. 1”, O. Lehto (e d.), Acad. Sci. Fenn., 1980, pp. 339-350

work page 1978
[6]

G. V. Chudnovsky, Contributions to the Theory of Transcende ntal Numbers, Math. Surveys and Monographs 19, Amer. Math. Soc., 1984

work page 1984
[7]

Essouabri

D. Essouabri. Singularit´ es de s´ eries de Dirichlet associ´ ees ` a des polynˆ omes de plusieurs variables et applications ` a la th´ eorie analytique des nombres. PhD The- sis, University Henri Poincar´ e-Nancy 1 (1995)

work page 1995
[8]

Essouabri

D. Essouabri. Singularit´ es de s´ eries de Dirichlet associ´ ees ` a des polynˆ omes de plusieurs variables et applications en th´ eorie analytiqu e des nombres. Ann. Inst. Fourier 47, No.2 (1997), 429–483

work page 1997

Show all 18 references

[9]

Essouabri and K

D. Essouabri and K. Matsumoto. Values at non-positive integers of generalized Euler-Zagier multiple zeta-functions . Acta Arith., to appear
[10]

Essouabri and K

D. Essouabri and K. Matsumoto. Values at non-positive integers of partially twisted multiple zeta-functions I . Comment. Math. Univ. St. Pauli, to appear
[11]

Friedman and A

E. Friedman and A. Pereira. Special values of Dirichlet series and zeta integrals . Int. J. Number Theory 8 (2012), 697–714

2012
[12]

Furusho, Y

H. Furusho, Y. Komori, K. Matsumoto and H. Tsumura. Desingularization of complex multiple zeta-functions. Amer. J. Math. 139 (2017), 147–173

2017
[13]

Y. Komori. An integral representation of multiple Hurwitz-Lerch zeta functions and generalized multiple Bernoulli numbers . Quart. J. Math. 61 (2010), 437–496

2010
[14]

Mathematics Unlimited — 2001 and Beyond

M. Kontsevich and D. Zagier. Periods. in “Mathematics Unlimited — 2001 and Beyond”, B. Engquist and W. Schmid (eds.), Springer, 2001, pp. 77 1–808

2001
[15]

K. Mahler. ¨Uber einen Satz von Mellin . Math. Ann. 100 (1928), no. 1, 384–398

1928
[16]

Matsumoto

K. Matsumoto. The analytic continuation and the asymptotic behaviour of c ertain multiple zeta-functions I . J. Number Theory 101 (2003), 223–243

2003
[17]

Number Theory for the Millennium II

K. Matsumoto. On the anayltic continuation of various multiple zeta-func tions. in “Number Theory for the Millennium II”, M. A. Bennett et al. (eds.), A K Peters, 2002, pp. 417–440. 41

2002
[18]

Num- ber Theoretic Methods, Future Trends

K. Matsumoto and L. Weng. Zeta-functions deﬁned by two polynomials . in “Num- ber Theoretic Methods, Future Trends”, S. Kanemitsu and C. Jia ( eds.), Kluwer Acad. Publ. 2002, pp. 233–262. Driss Essouabri Univ. Lyon, UJM-Saint-Etienne, CNRS, Institut Camille Jordan UMR 5208, Fa...

2002

Pith tools

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

[1] [1]

Arakawa, T

T. Arakawa, T. Ibukiyama and M. Kaneko, Bernoulli Numbers and Zeta Func- tions, Springer, 2014

work page 2014

[2] [2]

S´ eries de Dirichlet

Pierrette Cassou-Nogu` es. S´ eries de Dirichlet. Journ´ ees Arithm´ etiques de Metz (Metz, 1981), Ast´ erisque,94, Soc. Math. France, Paris, 1982, pp. 1–15

work page 1981

[3] [3]

Valeurs aux entiers n´ egatifs des s´ eries de Dirichlet associ´ ees ` a un polynˆ ome II.Amer

Pierrette Cassou-Nogu` es. Valeurs aux entiers n´ egatifs des s´ eries de Dirichlet associ´ ees ` a un polynˆ ome II.Amer. J. Math. 106 (1984), no. 2, 255–299. 40

work page 1984

[4] [4]

G. V. Chudnovsky. Algebraic independence of constants connected with expone n- tial and elliptic functions. Dokl. Akad. Nauk Ukrain. SSR Ser. A (1976), no. 8, 698-701, 767

work page 1976

[5] [5]

Proc. Intern. Congr. Math. Helsinki 1978, Vol. 1

G. V. Chudnovsky, Algebraic independence of values of exponential and ellipt ic functions. in “Proc. Intern. Congr. Math. Helsinki 1978, Vol. 1”, O. Lehto (e d.), Acad. Sci. Fenn., 1980, pp. 339-350

work page 1978

[6] [6]

G. V. Chudnovsky, Contributions to the Theory of Transcende ntal Numbers, Math. Surveys and Monographs 19, Amer. Math. Soc., 1984

work page 1984

[7] [7]

Essouabri

D. Essouabri. Singularit´ es de s´ eries de Dirichlet associ´ ees ` a des polynˆ omes de plusieurs variables et applications ` a la th´ eorie analytique des nombres. PhD The- sis, University Henri Poincar´ e-Nancy 1 (1995)

work page 1995

[8] [8]

Essouabri

D. Essouabri. Singularit´ es de s´ eries de Dirichlet associ´ ees ` a des polynˆ omes de plusieurs variables et applications en th´ eorie analytiqu e des nombres. Ann. Inst. Fourier 47, No.2 (1997), 429–483

work page 1997