Three-variable Mahler measures and special values of modular and Dirichlet $L$-series

TL;DR

This paper establishes explicit formulas connecting Mahler measures of specific Laurent polynomials to special values of modular and Dirichlet L-series, revealing deep links between algebraic, analytic, and hypergeometric structures.

Contribution

It proves that certain Mahler measures can be expressed as linear combinations of derivatives of modular and Dirichlet L-series, and derives new hypergeometric evaluations from these results.

Findings

Mahler measures expressed as linear combinations of L-series derivatives

New hypergeometric evaluations and transformations derived

Explicit formulas for Mahler measures of specific Laurent polynomials

Abstract

In this paper we prove that the Mahler measures of the Laurent polynomials $(x+x^{-1})(y+y^{-1})(z+z^{-1})+k$, $(x+x^{-1})^2(y+y^{-1})^2(1+z)^3z^{-2}-k$, and $x^4+y^4+z^4+1+k^{1/4}xyz$, for various values of $k$, are of the form $r_1 L'(f,0)+r_2 L'(\chi,-1)$, where $r_1,r_2\in \mathbb{Q}$, $f$ is a CM newform of weight 3, and $\chi$ is a quadratic character. Since it has been proved that these Maher measures can also be expressed in terms of logarithms and $_5F_4$-hypergeometric series, we obtain several new hypergeometric evaluations and transformations from these results.