Mahler measure, motivic regulators and Dirichlet $L$-values

TL;DR

This paper establishes a formula linking Mahler measures of certain polynomials to motivic regulators and special values of Dirichlet L-functions, advancing the understanding of their deep arithmetic connections.

Contribution

It derives a new formula connecting Mahler measures with motivic regulators and Dirichlet L-values, extending Deninger's work and analyzing Galois module structures.

Findings

Derived a formula relating Mahler measure to motivic regulators and Dirichlet L-values.

Under linear independence assumptions, obtained refined identities for specific L-values.

Connected Mahler measures of cyclotomic variants to motivic cohomology and special L-values.

Abstract

Inspired by the work of Deninger, we present a formula that relates the Mahler measure of a two-variable variant of cyclotomic polynomial to regulator of class in motivic cohomology associated to cyclotomic fields and linear combination of special values of the derivative of Dirichlet $L$-functions. The formula is derived by studying the Beilinson regulator map applied to systematically constructed elements in the motivic cohomology group. Under linear independence hypothesis on the derivative of partial Dirichlet $L$-values at $s=0$ and $-1$, we study a Galois module structure of the relevant motivic cohomology and obtain the refined identity for a single $L$-value.