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

TL;DR

This paper derives explicit formulas for the values of multiple zeta-functions with polynomial denominators at non-positive integers, revealing transcendental values and relations to Bernoulli numbers.

Contribution

It provides new explicit formulas for multiple zeta-functions at non-positive integers, including cases with polynomial denominators and power sums, involving period integrals.

Findings

Existence of trivial zeros for power sum denominators.

Some special values are transcendental.

Explicit formulas involve period integrals and Bernoulli number relations.

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.