The parametrized family of metric Mahler measures

Let M(α) denote the (logarithmic) Mahler measure of the algebraic number α. 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∈(0,∞], a metric version Mt of the Mahler measure, each having a triangle inequality of a different strength. We further examine the functions Mt, using them to present an equivalent form of Lehmerʼs conjecture. We show that the function t↦Mtt(α) is constructed piecewise from certain sums of exponential functions. We pose a conjecture that, if true, enables us to graph t↦Mt(α) for rational α.