# Multiple Stieltjes constants and Laurent type expansion of the multiple zeta functions at integer points

## TL;DR

This paper investigates the local behavior of multiple zeta functions at integer points, deriving a Laurent expansion with coefficients called multiple Stieltjes constants, which are obtained via a regularisation process.

## Contribution

It introduces multiple Stieltjes constants and provides a Laurent type expansion of multiple zeta functions at integer points, extending classical concepts to multiple variables.

## Key findings

- Laurent expansion of multiple zeta functions at integer points
- Definition of multiple Stieltjes constants via regularisation
- Expression of expansion in terms of smaller depth constants

## Abstract

In this article, we study the local behaviour of the multiple zeta functions at integer points and write down a Laurent type expansion of the multiple zeta functions around these points. Such an expansion involves a convergent power series whose coefficients are obtained by a regularisation process, similar to the one used in defining the classical Stieltjes constants for the Riemann zeta function. We therefore call these coefficients {\it multiple Stieltjes constants}. The remaining part of the above mentioned Laurent type expansion is then expressed in terms of the multiple Stieltjes constants arising in smaller depths.