A continuous version of multiple zeta functions and multiple zeta values

TL;DR

This paper introduces a continuous analogue of multiple zeta functions, exploring their analytic properties, algebraic structures, and explicit evaluations, expanding the understanding of multiple zeta values in a continuous setting.

Contribution

It defines and analyzes a new continuous version of multiple zeta functions, including their meromorphic continuation, algebraic relations, and explicit evaluations.

Findings

Continuous multiple zeta functions have meromorphic continuation with simple poles.

They satisfy the shuffle product structure.

Explicit calculations relate them to multiple polylogarithms.

Abstract

In this paper we define a continuous version of multiple zeta functions. They can be analytically continued to meromorphic functions on $\mathbb{C}^r$ with only simple poles at some special hyperplanes. The evaluations of these functions at positive integers (continuous multiple zeta values) satisfy the shuffle product. We give a detailed analysis about the depth structure of continuous multiple zeta values. There are also sum formulas for continuous multiple zeta values. Lastly we calculate some special continuous multiple zeta values in terms of special values of multiple polylogarithms.