Remark on a symmetric zeta function

TL;DR

This paper introduces a symmetric zeta function, explores its analytic continuation and pole structure, computes a specific residue, and analyzes a divergent series related to the Euler constant.

Contribution

It defines a new symmetric zeta function, studies its meromorphic extension, and connects a divergent series to the Euler constant, providing new insights into multiple zeta functions.

Findings

Symmetric zeta function can be meromorphically extended to a7a3^3 with simple poles.

Calculated the multiple residue at a special point.

Analyzed growth of a divergent series related to the symmetric zeta function and linked it to the Euler constant.

Abstract

In this paper we define a symmetric zeta function. We show that it can be analytically continued to a meromorphic function on $\mathbb{C}^3$ with only simple poles at some special hyperplanes. We also calculate the value of a multiple residue at one special point. For a divergent multiple series, which can be viewed as the value of the symmetric zeta function at the point $(1,1,1)$, we give a very detailed analysis on its growth and we relate it to the classical Euler constant.