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

TL;DR

This paper establishes the meromorphic continuation of multiple zeta functions across their entire domain and analyzes their asymptotic behavior at non-positive integers, advancing understanding of their complex properties.

Contribution

It provides the first comprehensive proofs of meromorphic continuation and asymptotic behavior of multiple zeta-functions at non-positive integers.

Findings

Meromorphic continuation of multiple zeta functions to the whole space

Asymptotic formulas near non-positive integers

New theorems establishing analytic properties

Abstract

We prove two theorems. Theorem 1 gives the meromorphic continuation of the multiple zeta function to the whole space. In Theorem 2, we prove asymptotic behavior near the non-positive integers.