Relating Mahler measures and Dirichlet $L$-values: new evidence for Chinburg's conjectures

TL;DR

This paper provides new numerical evidence supporting Chinburg's conjectures relating Mahler measures of polynomials to derivatives of Dirichlet L-functions, and proves the weak form with cyclotomic coefficients.

Contribution

It doubles the known numerical instances of Chinburg's conjectures and offers an explicit approach based on Boyd and Rodriguez-Villegas's work.

Findings

8 new instances of the strong conjecture identified

18 new instances of the weak conjecture identified

Proved the weak conjecture with cyclotomic coefficients

Abstract

Let $\chi_{-f}$ be the odd quadratic Dirichlet character of conductor $f$, and let $\mathrm{m}(P)$ denote the Mahler measure of a polynomial $P$. In 1984, Chinburg conjectured that for any such $\chi_{-f}$ there exist an integral bivariate rational function $P$ (and, in the strong form, an integral polynomial) such that $\mathrm{m}(P)$ is a rational multiple of $L'(\chi_{-f},-1)$. The strong form of the conjecture was previously known to hold for $18$ values of $f$. We double the number of numerical examples, giving $8$ new instances of the strong and $18$ new instances of the weak conjecture. Our examples arise from an explicit approach, which also captures almost all of the previously known results, and is based on work of Boyd and Rodriguez-Villegas. Moreover, we prove Chinburg's weak conjecture if we allow cyclotomic coefficients.