Number Theory Seminar: The parametrized family of metric Mahler measures

Let $M(\alpha)$ denote the Mahler measure of the algebraic number $\alpha$. Dubickas and Smyth constructed a modified version $M_1$ of $M$ having the triangle inequality. $M_1$ is called the metric Mahler measure. We produce an entire parametrized family $\{M_t\}$ of metric Mahler measures which gives rise to a new reformulation of Lehmer's problem. We further examine the functions $t\mapsto M_t(\alpha)$, for fixed $\alpha$, showing that they are constructed piecewise from certain simpler functions.