Integrals Over Polytopes, Multiple Zeta Values and Polylogarithms, and Euler's Constant

TL;DR

This paper investigates integrals over a specific triangle, expressing them via multiple zeta values and polynomials, deriving asymptotics, and exploring connections to Euler's constant and higher-dimensional polylogarithm integrals.

Contribution

It provides new explicit formulas for integrals over polytopes in terms of multiple zeta values and extends these results to higher-dimensional cases involving multiple polylogarithms.

Findings

Expressions for integrals as linear combinations of multiple zeta values.

Asymptotic expansions of integrals and zeta sums.

Relations between multiple polylogarithms and zeta values.

Abstract

Let $T$ be the triangle with vertices (1,0), (0,1), (1,1). We study certain integrals over $T$, one of which was computed by Euler. We give expressions for them both as a linear combination of multiple zeta values, and as a polynomial in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler's constant. In the final section, we evaluate more general integrals -- one is a Chen (Drinfeld-Kontsevich) iterated integral -- over some polytopes that are higher-dimensional analogs of $T$. This leads to a relation between certain multiple polylogarithm values and multiple zeta values.