Log-sine evaluations of Mahler measures

TL;DR

This paper evaluates complex Mahler measures using log-sine integrals, linking them to zeta and polylogarithmic values, and develops methods for broader applications.

Contribution

It introduces new techniques for expressing Mahler measures in terms of log-sine integrals and extends these methods to additional families of measures.

Findings

Evaluated several higher and multiple Mahler measures.

Connected Mahler measures to zeta and polylogarithmic values.

Developed a framework for future Mahler measure evaluations.

Abstract

We provide evaluations of several recently studied higher and multiple Mahler measures using log-sine integrals. This is complemented with an analysis of generating functions and identities for log-sine integrals which allows the evaluations to be expressed in terms of zeta values or more general polylogarithmic terms. The machinery developed is then applied to evaluation of further families of multiple Mahler measures.