The parametrized family of metric Mahler measures

TL;DR

This paper introduces a family of metric Mahler measures parametrized by t, explores their properties, and presents an equivalent form of Lehmer's conjecture, revealing piecewise exponential structures in these functions.

Contribution

The author generalizes metric Mahler measures to a continuous family indexed by t, and connects these measures to Lehmer's conjecture through new functional representations.

Findings

The functions M_t(α) are piecewise sums of exponential functions.

A new equivalent form of Lehmer's conjecture is proposed.

Graphical representations of M_t(α) for rational α are suggested.

Abstract

Let $M(\alpha)$ denote the (logarithmic) Mahler measure of the algebraic number $\alpha$. Dubickas and Smyth, and later Fili and the author, examined metric versions of $M$. The author generalized these constructions in order to associate, to each point in $t\in (0,\infty]$, a metric version $M_t$ of the Mahler measure, each having a triangle inequality of a different strength. We further examine the functions $M_t$, using them to present an equivalent form of Lehmer's conjecture. We show that the function $t\mapsto M_t(\alpha)^t$ is constructed piecewise from certain sums of exponential functions. We pose a conjecture that, if true, enables us to graph $t\mapsto M_t(\alpha)$ for rational $\alpha$.