标题： 关于Bessel函数的积分及其在Mahler测度中的应用（附录由J.S. Friedman*提供） 显示英文标题

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

摘要： 在最近的一篇论文中，Cogdell、Jorgenson 和 Smajlović 团队为复线性形式的对数 Mahler 测度建立了无限级数表示，该形式包含 4 个或更多变量。 我们通过一个积分进行界定，该积分的被积函数涉及随机游走概率密度$a\displaystyle\int_0^\infty tJ_0(at) \displaystyle\prod_{m=0}^2 J_0(r_m t)dt$，其中$J_0$是第一类零阶贝塞尔函数，$a$和{$r_m$}是正实数，从而确立了 3 个变量的情况。 为了便于证明，我们开发了积分在其已知发散点处渐近行为的另一种描述。 作为计算辅助以适应数值实验，附录中提出了一个计算这些级数的算法。 显示英文摘要

摘要： 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.