The Mahler measure of a family of polynomials with arbitrarily many variables

TL;DR

This paper derives an exact formula for the Mahler measure of an infinite family of multivariable polynomials, linking it to polylogarithms, zeta functions, and Dirichlet L-functions.

Contribution

It introduces a novel method to compute Mahler measures for complex polynomial families using integral transformations and polylogarithm evaluations.

Findings

Exact formula for Mahler measure involving polylogarithms at roots of unity

Connections established between Mahler measures and special values of zeta and L-functions

Method applicable to polynomials with arbitrarily many variables

Abstract

We present an exact formula for the Mahler measure of an infinite family of polynomials with arbitrarily many variables. The formula is obtained by manipulating the integral defining the Mahler measure using certain transformations, followed by an iterative process that reduces this computation to the evaluation of certain polylogarithm functions at sixth roots of unity. This yields values of the Riemann zeta function and the Dirichlet $L$-function associated to the character of conductor 3.