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

TL;DR

This paper proves a new integral representation related to Mahler measure for three variables, extending previous work for four or more variables, and provides computational tools for numerical evaluation.

Contribution

It establishes the case of three variables for a Mahler measure integral, developing asymptotic analysis and an algorithm for series computation.

Findings

Bounded an integral involving Bessel functions for three variables.

Developed an alternative asymptotic description of the integral.

Provided an algorithm for numerical series calculation.

Abstract

In a recent paper the team of Cogdell, Jorgenson and Smajlovi\'c 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 $a\displaystyle\int_0^\infty tJ_0(at) \displaystyle\prod_{m=0}^2 J_0(r_m t)dt$, 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.