On Mordell-Tornheim sums and multiple zeta values

TL;DR

This paper demonstrates that Mordell-Tornheim sums with positive integer arguments can be expressed as rational combinations of multiple zeta values, revealing new relationships between these special sums and multiple zeta values.

Contribution

It establishes explicit relations between Mordell-Tornheim sums and multiple zeta values, extending the understanding of their algebraic structure and parity properties.

Findings

Mordell-Tornheim sums with positive integers are expressible via multiple zeta values.

Sums with opposite parity weight and depth can be decomposed into products of lower-depth multiple zeta values.

Provides a framework for expressing complex sums in terms of well-studied multiple zeta values.

Abstract

We prove that any Mordell-Tornheim sum with positive integer arguments can be expressed as a rational linear combination of multiple zeta values of the same weight and depth. By a result of Tsumura, it follows that any Mordell-Tornheim sum with weight and depth of opposite parity can be expressed as a rational linear combination of products of multiple zeta values of lower depth.