Analytic properties of generalized Mordell-Tornheim type of multiple zeta-functions and L-functions

TL;DR

This paper investigates the analytic properties of generalized multiple zeta-functions and L-functions, extending known types by proving their meromorphic continuation and analyzing singularities in multi-dimensional complex spaces.

Contribution

It introduces a unified framework for generalized multiple zeta-functions and L-functions, including Euler-Zagier and Mordell-Tornheim types, and establishes their meromorphic continuation.

Findings

Proved meromorphic continuation to multi-dimensional complex space

Identified possible singularities of the generalized functions

Unified treatment of multiple zeta and L-functions

Abstract

Analytic properties of three types of multiple zeta functions, that is, the Euler-Zagier type, the Mordell-Tornheim type and the Apostol-Vu type have been studied by a lot of authors. In particular, in the study of multiple zeta functions of the Apostol-Vu type, a generalized multiple zeta function, including both the Euler-Zagier type and the Apostol-Vu type, was introduced.In this paper, similarly we consider generalized multiple zeta-functions and $ L $-functions, which include both the Euler-Zagier type and the Mordell-Tornheim type as special cases.We prove the meromorphic continuation to the multi-dimensional complex space, and give the results on possible singularities.