On an integral of J-Bessel functions and its application to Mahler measure (with an appendix by J.S. Friedman*)

Abstract

In a recent paper the team of Cogdell, Jorgenson and Smajlović develop infinite series representations for the logarithmic Mahler measure of a complex linear form, with 4 or more variables. We establish the case of 3 variables, by bounding an integral with integrand involving the random walk probability density $a{\int }_0^{\infty }t{J}_0(at)\prod _{m=0}^2{J}_0(r_{m}t)dt$, where $J_0$ is the order zero Bessel function of the first kind, and $a$ and {$r_m$} are positive real numbers. To facilitate our proof we develop an alternative description of the integral's asymptotic behavior at its known points of divergence. As a computational aid to accommodate numerical experiments, an algorithm to calculate these series is presented in the Appendix.

To appear in the Bulletin of the Australian Mathematical Society; 12 pages, 4 figures

References in corpus (1)

• Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space