Mahler measure of P d polynomials

This article investigates the Mahler measure of a family of 2-variate polynomials, denoted by P d , for d≥1, unbounded in both degree and genus. By using a closed formula for the Mahler measure 13, we are able to compute m(P d ), for arbitrary d, as a sum of the values of dilogarithm at special roots of unity. We prove that m(P d ) converges, and the limit is proportional to ζ(3), where ζ is the Riemann zeta function. The proof we give is computational and based on the estimation of the error of Riemann sums of a bivariate function. We describe a second possible shorter proof based on a conjectural generalization of the theorem of Boyd–Lawton and a result of D’Andrea and Lalín 11.