Values of multiple zeta functions with polynomial denominators at non-positive integers
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Publication:4993871
DOI10.1142/S0129167X21500385zbMATH Open1470.11226arXiv1908.09248MaRDI QIDQ4993871
Author name not available (Why is that?)
Publication date: 11 June 2021
Published in: (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/1908.09248
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