The distribution of Mahler's measures of reciprocal polynomials

Abstract

We study the distribution of Mahler's measures of reciprocal polynomials with complex coefficients and bounded even degree. We discover that the distribution function associated to Mahler's measure restricted to monic reciprocal polynomials is a reciprocal (or antireciprocal) Laurent polynomial on [1,∞) and identically zero on [0,1). Moreover, the coefficients of this Laurent polynomial are rational numbers times a power of π. We are led to this discovery by the computation of the Mellin transform of the distribution function. This Mellin transform is an even (or odd) rational function with poles at small integers and residues that are rational numbers times a power of π. We also use this Mellin transform to show that the volume of the set of reciprocal polynomials with complex coefficients, bounded degree, and Mahler's measure less than or equal to one is a rational number times a power of π.