Instant Multiple Zeta Values at Non-Positive Integers and Bernoulli Functions

TL;DR

This paper provides an elementary method for evaluating multiple Zeta functions at non-positive integers, explores the Fourier theory of Bernoulli polynomial products, and shows the equivalence of polynomial expressions for Hurwitz Zeta and Bernoulli polynomials.

Contribution

It introduces an elementary approach to evaluate multiple Zeta functions at non-positive integers and establishes the equivalence between Hurwitz Zeta and Bernoulli polynomial expressions.

Findings

Elementary evaluation method for multiple Zeta functions at non-positive integers

Fourier analysis of Bernoulli polynomial products

Equivalence of polynomial expressions for Hurwitz Zeta and Bernoulli polynomials

Abstract

We give an instant evaluation of multiple Zeta function at non-positive integers by elementary methods and discuss the Fourier theory (on unit interval) of the product of Bernoulli polynomials.We also show that the polynomial expression for Hurwitz Zeta function(at non-positive integral values of the first variable) and the polynomial expression for Bernoulli polynomials are equivalent.