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arXiv · 2012.04165

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

George Anton·Jessen A. Malathu·Shelby Stinson
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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.

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George Anton, Jessen A. Malathu, Shelby Stinson. 2020-12-08. On an integral of J-Bessel functions and its application to Mahler measure (with an appendix by J.S. Friedman*). https://arxiv.org/abs/2012.04165

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