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

George Anton, Jessen A. Malathu, Shelby Stinson
2021 arXiv   pre-print
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∫_0^∞ tJ_0(at) ∏_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
more » ... tive 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.
arXiv:2012.04165v3 fatcat:l6zi6mgi3zambpqgfvbp3cc5oi