The parametrized family of metric Mahler measures

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 $t\in (0,\mathrm{\infty }]$, 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 $\alpha$.