ON AN INTEGRAL OF -BESSEL FUNCTIONS AND ITS APPLICATION TO MAHLER MEASURE

Abstract

\n Cogdell et al. [‘Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space’, Trans. Amer. Math. Soc. (2021), to appear] developed infinite series representations for the logarithmic Mahler measure of a complex linear form with four or more variables. We establish the case of three variables by bounding an integral with integrand involving the random walk probability density \n \n \n $a\\int _0^\\infty tJ_0(at) \\prod _{m=0}^2 J_0(r_m t)\\,dt$\n \n , where \n \n \n $J_0$\n \n is the order-zero Bessel function of the first kind and a and \n \n \n $r_m$\n \n are positive real numbers. To facilitate our proof we develop an alternative description of the integral’s asymptotic behaviour at its known points of divergence. As a computational aid for numerical experiments, an algorithm to calculate these series is presented in the appendix.