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Values at non-positive integers of partially twisted multiple zeta-functions II

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Pith's one-line read This paper proves an explicit formula for the values at non-positive integer points of partially twisted multiple zeta-functions with general polynomial denominators, expressing them through de Crisenoy's fully twisted zeta values at…

desk verdict A solid, genuinely new reduction of the partially twisted multiple zeta case with general polynomial denominators to de Crisenoy's fully twisted theorem; I found no fatal flaw. read the letter →

arxiv 2506.20150 v1 pith:DHMNHVJL submitted 2025-06-25 math.NT

Driss Essouabri , Kohji Matsumoto , Simon Rutard This is my paper · ORCID
classification math.NT MSC 11M3211M3511M41
keywords partiallytwistedmultiplezeta-functionspolynomialdenominatorsMellin-Barnesintegralformulavaluesatnon-positiveintegersLerchzeta-functionBernoullinumbersmeromorphiccontinuationtranscendence
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

Partially twisted multiple zeta-functions have twisting characters in all summation variables except the last, and denominators that are products of polynomial factors. This paper establishes that for general polynomial denominators satisfying the HDF positivity and decay condition, every non-positive integer tuple is a regular point, and the value there is given by an explicit finite formula. The formula expresses the value in terms of de Crisenoy's fully twisted zeta values at integers, twisted Bernoulli numbers, and multinomial coefficients. A consequence visible in the paper's examples is that these values can be transcendental, such as non-constant rational polynomials in pi. The proof works by a Mellin-Barnes separation of the last summation variable and iterative contour shifts, resting on a new moderate-growth estimate for the fully twisted zeta-function.

What carries the argument

The load-bearing mechanism is the multiple Mellin-Barnes integral formula (Proposition 8), applied to the inner sum over m_n so that the argument of the resulting zeta factor depends on new integration variables z. Iterative left shifts of the contour in the variables z_j produce the auxiliary integrals $I^{{(r)}}$ and $J^{{(r)}}$ (Definitions 28-29), whose residue calculus reduces everything to values of de Crisenoy's fully twisted zeta-function at integers. All shifts are justified by Proposition 6, the new proof that the fully twisted zeta-function is of moderate growth; the uniform estimate behind it is Proposition 15.

What would settle it

Set n=T=1 and P(X)=1+X in the formula of Section 8.2; since zeta_1(s;P;1) equals zeta(s)-1 for this P, comparing the right-hand side at s=-N with the known value zeta(-N)=-B_{N+1}/(N+1) at any N>=0 would immediately detect an error in Theorem 24.

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Extended reading notes

Core claim

The central claim is Theorem 24: under the HDF and growth hypotheses on the polynomials P_1,...,P_T, for every N=(N_1,...,N_T) with non-negative integer entries the special value zeta_n(-N;P;mu_{n-1}) equals the right-hand side of formula (6.2), a finite sum of products of multinomial coefficients, twisted Bernoulli numbers, and values of the fully twisted zeta-function $zeta^{{dC}}$_{n-1} at integer points. Corollary 23 states that zeta_n(s;P;mu_{n-1}) continues meromorphically to all of C^T with singularities contained in explicit hyperplanes, and none of the non-positive integer points is singular. The same Mellin-Barnes mechanism yields the intermediate d=1 and d=2 cases (Theorems 17 and 18) and recovers the expected linear-case formulas.

Load-bearing premise

The argument depends on the new moderate-growth estimate for de Crisenoy's fully twisted zeta-function; if that estimate failed for some polynomial denominator satisfying the paper's HDF positivity and decay condition, the contour shifts that produce the explicit formula would not be justified.

Editorial extensions

If this is right

  • Every non-positive integer point is a regular point of zeta_n(s;P;mu_{n-1}), so no limiting procedure is needed at those points.
  • The special values are completely determined by fully twisted zeta values at integers, twisted Bernoulli numbers, and multinomial coefficients, so no new transcendental constants enter beyond those already present in the fully twisted case.
  • When the leading coefficient of P_T is constant, the special values lie in the field generated over Q by the roots of unity mu_j and the coefficients of the polynomials, meaning no new transcendence appears.
  • In the example P_1=1, P_2=1+X_2+X_1^q X_2^2 with mu=-1 and q even, the values are non-constant rational polynomials in pi and are therefore transcendental.
  • The authors state that the same method should extend inductively to cases with fewer twists (k <= n-2), in the style of their earlier linear and power-sum treatment.

Reading between the lines

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

  • A direct test of the formula in the one-variable case P(X)=1+X, where zeta_1(s;P;1) equals zeta(s)-1, would verify the residue calculus in a clean setting without needing any numerical analytic continuation.
  • If Proposition 6's moderate-growth bound can be made explicit with constants, the Mellin-Barnes representation (7.1) could yield uniform estimates for these zeta-functions in vertical strips, which would be useful for zero or value-distribution questions.
  • The appearance of fully twisted values at rational arguments such as (a_1 k_2+1)/a_2 in the d=2 case suggests a broader pattern: partial twisting may force special values to be governed by Lerch-type values at rational points, linking to polylogarithms at roots of unity.
  • The same separation-of-variables trick should adapt to partial twists with more than one untwisted factor, provided the moderate-growth estimate holds for the corresponding fully twisted function with those extra variables.
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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies the partially twisted multiple zeta-function ζ_n(s;P;µ_{n-1}) with general polynomial denominators, in the case where all but one of the summation variables carry twists. Using the multiple Mellin-Barnes formula, the authors reduce the problem to the fully twisted case solved by de Crisenoy. The main results are Theorem 22 and Corollary 23, giving meromorphic continuation to C^T and a description of the singular hyperplanes, and Theorem 24, which gives an explicit formula for the values at non-positive integer points in terms of de Crisenoy's fully twisted values, twisted Bernoulli numbers, and multinomial coefficients. The proof requires a new moderate-growth estimate for de Crisenoy's fully twisted zeta functions, proved in Section 3 as Proposition 6. Several worked examples show the formula in action, including an example where the special values are transcendental.

Significance. The result is a substantial extension of the authors' Part I, where only linear and power-sum denominators were treated. The explicit special-value formula (6.2) is new and is expressed in a usable form, and the companion moderate-growth result (Proposition 6) is a useful complement to de Crisenoy's theorem. The transcendental example (Example 35) is particularly valuable because it shows that the arithmetic nature of the values changes genuinely with the shape of the polynomial denominator. The paper is carefully written and the main derivation is internally consistent; no circularity or fitted parameters are present.

minor comments (4)
  1. [Section 3.2, proof of Proposition 15] The sentence 'Since at each stage of his recurrence, de Crisenoy proceeds only by integration by parts, it is clear that his proof implies...' compresses the uniformity argument in a way that is load-bearing for Proposition 6. Please spell out explicitly how the induction constants remain uniform in the functions f_k (and not merely in the data Q, R_t), and why the coefficients produced by the recurrence have at most polynomial growth in s. As written, this is the only place where an unstated uniformity is required.
  2. [Section 7.4, proof of Theorem 24] The displayed limit evaluates Γ(eg(s_T;-ℓ))/Γ(s_T), but the term being analyzed contains the product Γ(eg(s_T;-ℓ))Γ(g(s_T;-ℓ))/Γ(s_T). Please add an explicit sentence explaining that Γ(g(-N_T;-ℓ)) = (|ℓ|+i-N_T-1)! and that the product limit is what produces the second sum in (6.2). Without this clarification, the transition from the limit to the final formula is not fully visible.
  3. [Section 6] When applying de Crisenoy's Proposition 3 to the fully twisted family P(Q) = (P_1,...,P_{T-1},Q_0,...,Q_d), the hypotheses require the product of all entries to tend to infinity. The authors assume (2.4) instead. Please add a sentence explaining that (2.4) implies the needed growth condition, e.g. by noting that each positive polynomial has positive infimum on [1,∞)^{n-1}.
  4. [Throughout] There are a few small typographical slips: a double comma in the definition of P_{n1,n2,n3}^ε in Section 3.3, and a missing closing parenthesis in the first line of Example 35. These should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central value formula (6.2) is an honest Mellin-Barnes reduction to de Crisenoy's external fully twisted theorem plus a newly proved moderate-growth estimate.

full rationale

The derivation chain relevant to circularity is self-contained. The main formula (6.2) is obtained by writing zeta_n(s;P;mu_{n-1}) as a multiple Mellin-Barnes integral (7.1), shifting contours, collecting residues from the gamma factors and the factor (a_0 s_T + <alpha,z> - 1), and then evaluating at s = -N. The only external black box is de Crisenoy's Proposition 3, a published independent theorem on fully twisted zeta functions; the paper's own analytic input is Proposition 6, the moderate-growth estimate, which is proved in Section 3 by adapting de Crisenoy's integration-by-parts recurrence. That estimate is used to justify the contour shifts and to control remainder integrals; it does not assume the final value formula. No parameter is fitted to the target values, and the final expression reduces the partially twisted values to fully twisted special values, twisted Bernoulli numbers, and multinomial coefficients, which are distinct objects. The authors' earlier paper [9] is cited for context and motivation, not as a load-bearing input to Theorem 24. The transcendence example derives from the classical transcendence of pi, not from a circular redefinition. The paper candidly notes that some fully twisted input values at positive arguments remain 'rather mysterious', but this affects the explicitness of the formula, not the circularity of the derivation.

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

No fitting to data and no invented entities. The proof depends on standard analytic tools and on explicit hypotheses on the polynomials; the final formulas introduce no free constants. The only external mathematical input is de Crisenoy's theorem and classical Mellin-Barnes theory.

assumptions (5)
  • domain assumption HDF condition on P_1,...,P_{T-1},Q_0,...,Q_d and the growth condition (2.4)
    Stated in Section 6; required for de Crisenoy's theorem, absolute convergence, and the Mellin-Barnes separation of variables.
  • domain assumption de Crisenoy's theorem: under HDF, fully twisted series is entire and its non-positive integer values are given by formula (2.3)
    External theorem quoted from reference [4]; the paper's reduction is built on it.
  • standard math Mellin-Barnes integral representations (Propositions 7 and 8)
    Classical integral identities used to separate m_n from the other summation variables.
  • standard math Residue theorem and contour shifts are valid under the moderate-growth and gamma-decay conditions
    Used throughout Sections 4 through 7; justified by Proposition 6 and Stirling estimates for the gamma function.
  • standard math Classical special value formula zeta(-m) = -B_{m+1}/(m+1) and the functional equation for the gamma function
    Used in Section 7.4 to evaluate limits at non-positive integers.

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Pith. "Pith review of Values at non-positive integers of partially twisted multiple zeta-functions II." pith.science (2026). https://pith.science/paper/DHMNHVJL

@misc{pith2026250620150,
  author       = {Pith},
  title        = {Pith review of: Values at non-positive integers of partially twisted multiple zeta-functions II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DHMNHVJL}},
  note         = {Machine review of arXiv:2506.20150}
}
read the original abstract

We study the values at non-positive integer points of multi-variable twisted multiple zeta-functions, whose each factor of the denominator is given by polynomials. The fully twisted case was already answered by de Crisenoy. On the partially twisted case, in one of our former article we studied the case when each factor of the denominator is given by linear forms or power-sum forms. In the present paper we treat the case of general polynomial denominators, and obtain explicit forms of the values at non-positive integer points. Our strategy is to reduce to the theorem of de Crisenoy for the fully twisted case, via the multiple Mellin-Barnes integral formula. We observe that in some cases the obtained values are transcendental.

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Works this paper leans on

23 extracted references · 22 canonical work pages · cited by 1 Pith paper

  1. [1]

    Akiyama, S

    S. Akiyama, S. Egami and Y. Tanigawa, Analytic continuation of multi- ple zeta-functions and their values at non-positive integers, Acta Arith. 98(2001), 107–116

    work page 2001

  2. [2]

    Choie and K

    Y. Choie and K. Matsumoto, Functional equations for double series of Euler type with coefficients, Adv. Math.292(2016), 529–557

    work page 2016

  3. [3]

    Choie and K

    Y. Choie and K. Matsumoto, Functional equations for double series of Euler-Hurwitz-Barnes type with coefficients, in RIMS Kˆ okyˆ uroku BessatsuB68(2017), 91–109

    work page 2017

  4. [4]

    de Crisenoy, Values atT-tuples of negative integers of twisted multi- variable zeta series associated to polynomials of several variables, Com- positio Math.142(2006), no

    M. de Crisenoy, Values atT-tuples of negative integers of twisted multi- variable zeta series associated to polynomials of several variables, Com- positio Math.142(2006), no. 6, 1373–1402. 40

    work page 2006

  5. [5]

    de Crisenoy and D

    M. de Crisenoy and D. Essouabri, Relations between values atT-tuples of negative integers of twisted multivariable zeta series associated to polynomials of several variables, J. Math. Soc. Japan60(2008), 1–16

    work page 2008

  6. [6]

    Essouabri, Singularit´ es de s´ eries de Dirichlet associ´ ees ` a des polynˆ omes de plusieurs variables et application en th´ eorie analytique des nombres, Ann

    D. Essouabri, Singularit´ es de s´ eries de Dirichlet associ´ ees ` a des polynˆ omes de plusieurs variables et application en th´ eorie analytique des nombres, Ann. Inst. Fourier47(1997), 429–483

    work page 1997

  7. [7]

    Essouabri, Zeta functions associated to Pascal’s triangle mod p, Japanese J

    D. Essouabri, Zeta functions associated to Pascal’s triangle mod p, Japanese J. Math.31(2005), 157–174

    work page 2005

  8. [8]

    Recent Trends on Zeta functions in Algebra and Geometry

    D. Essouabri, Height zeta functions on generalized projective toric va- rieties, in “Recent Trends on Zeta functions in Algebra and Geometry”, Contemp. Math., Amer. Math. Soc. and Real Soc. Mat. Espanol., 2012, 34pp

    work page 2012

Show all 23 references
  1. [9]

    Essouabri and K

    D. Essouabri and K. Matsumoto, Values at non-positive integers of partially twisted multiple zeta-functions I, Comment. Math. Univ. St. Pauli67(2019), no. 1, 83–100

    2019

  2. [10]

    Essouabri and K

    D. Essouabri and K. Matsumoto, Values at non-positive integers of gen- eralized Euler-Zagier multiple zeta-functions, Acta Arith.193(2020), 109–131

    2020

  3. [11]

    Essouabri and K

    D. Essouabri and K. Matsumoto, Values of multiple zeta functions with polynomial denominators at non-positive integers, Intern. J. Math.32 (2021), 2150038, 41pp

    2021

  4. [12]

    Komori, An integral representation of multiple Hurwitz-Lerch zeta functions and generalized multiple Bernoulli numbers, Quart

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

    2010

  5. [13]

    Komori, K

    Y. Komori, K. Matsumoto and H. Tsumura, Functional equations for doubleL-functions and values at non-positive integers, Intern. J. Num- ber Theory7(2011), 1441–1461

    2011

  6. [14]

    Matsumoto, Functional equations for double zeta-functions, Math

    K. Matsumoto, Functional equations for double zeta-functions, Math. Proc. Cambridge Phil. Soc.136(2004), 1–7

    2004

  7. [15]

    Matsumoto, T

    K. Matsumoto, T. Onozuka and I. Wakabayashi, Laurent series expan- sions of multiple zeta-functions of Euler-Zagier type at integer points, Math. Z.295(2020), 623–642

    2020

  8. [16]

    Mellin, Eine Formel f¨ ur den Logarithmus transcendenter Funktionen von endlichem Geschlecht, Acta Soc

    H. Mellin, Eine Formel f¨ ur den Logarithmus transcendenter Funktionen von endlichem Geschlecht, Acta Soc. Sci. Fenn.29, no. 4 (1900), 3–49. 41

    1900

  9. [17]

    Murahara and T

    H. Murahara and T. Onozuka, Asymptotic behavior of the Hurwitz- Lerch multiple zeta functions at non-positive integer points, Acta Arith. 205(2022), 191–210

    2022

  10. [18]

    Onozuka, Analytic continuation of multiple zeta-functions and the asymptotic behavior at non-positive integers, Funct

    T. Onozuka, Analytic continuation of multiple zeta-functions and the asymptotic behavior at non-positive integers, Funct. Approx. Com- ment. Math.49(2013), 331–348

    2013

  11. [19]

    Rutard, Values and derivative values at nonpositive integers of gen- eralized multiple Hurwitz zeta functions, to appear in J

    S. Rutard, Values and derivative values at nonpositive integers of gen- eralized multiple Hurwitz zeta functions, to appear in J. Math. Soc. Japan, arXiv:2312.04725

    arXiv

  12. [20]

    Saha, Multiple Stieltjes constants and Laurent type expansion of the multiple zeta functions at integer points, Selecta Math.28(2022), paper no

    B. Saha, Multiple Stieltjes constants and Laurent type expansion of the multiple zeta functions at integer points, Selecta Math.28(2022), paper no. 6, 41pp

    2022

  13. [21]

    C. Salinas Zavala, Multivariable zeta functions, multivariable taube- rian theorems and applications, PhD thesis (2021), Universit´ e de Lyon; Universit´ e nationale d’ing´ enierie (Lima)

    2021

  14. [22]

    Sasaki, Multiple zeta values for coordinatewise limits at non-positive integers, Acta Arith.136(2009), 299–317

    Y. Sasaki, Multiple zeta values for coordinatewise limits at non-positive integers, Acta Arith.136(2009), 299–317

    2009

  15. [23]

    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge Univ. Press, Cambridge, 1927. D. Essouabri: Universit´ e Jean Monnet, Institut Camille Jordan (CNRS, UMR 5208), 20 rue du Docteur R´ emy Annino, Saint-´Etienne 42000, France. E-mail address: dris...

    1927

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