Higher Mahler measures and zeta functions

TL;DR

This paper generalizes the Mahler measure by integrating higher powers of logarithms and explores zeta Mahler measures, connecting them to special values of zeta functions, Dirichlet L-functions, and polylogarithms.

Contribution

It introduces a generalized framework for Mahler measures involving higher powers and zeta measures, providing explicit computations and linking to special functions.

Findings

Computed specific examples of higher Mahler measures.

Established connections to special values of zeta and L-functions.

Linked generalized measures to polylogarithms.

Abstract

We consider a generalization of the Mahler measure of a multivariable polynomial $P$ as the integral of $\log^k|P|$ in the unit torus, as opposed to the classical definition with the integral of $\log|P|$. A zeta Mahler measure, involving the integral of $|P|^s$, is also considered. Specific examples are computed, yielding special values of zeta functions, Dirichlet $L$-functions, and polylogarithms.