Cogdell et al. [‘Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space’, Trans. Amer. Math. Soc. (2021), to appear] developed infinite series representations for the logarithmic Mahler measure of a complex linear form with four or more variables. We establish the case of three variables by bounding an integral with integrand involving the random walk probability density $a\int _0^\infty tJ_0(at) \prod _{m=0}^2 J_0(r_m t)\,dt$ , 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 behaviour at its known points of divergence. As a computational aid for numerical experiments, an algorithm to calculate these series is presented in the appendix.

科格德尔等人。['通过复射影空间上的克罗内克极限公式评估线性形式的马勒测度'，Trans。阿米尔。数学。社会党。（2021），出现] 为具有四个或更多变量的复线性形式的对数马勒测度开发了无限级数表示。我们通过用涉及随机游走概率密度 $a\int _0^\infty tJ_0(at) \prod _{m=0}^2 J_0(r_m t)\,dt$ 的被积函数来确定三个变量的情况，其中 $J_0$ 是第一类零阶贝塞尔函数，a和 $r_m$ 是正实数。为了便于我们的证明，我们开发了积分在已知发散点处的渐近行为的替代描述。作为数值实验的计算辅助工具，附录中提供了计算这些级数的算法。