Integral Representations and Asymptotics for a Family of Areal Mahler Measures

Abstract

We study a problem posed by Matilde Lalín concerning the areal Mahler measures of the multivariable polynomial family $$P_m(x_1,\dots ,x_m,u)=\prod _{j=1}^m(1+x_j)+u\prod _{j=1}^m(1-x_j),m\ge 1.$$ Using a probabilistic reformulation, we derive convolution and one-dimensional Fourier integral representations for ${\mathrm{m}}_{\mathbb{D}}(P_m)$. We prove that, for every fixed $m$, the value ${\pi }^m{\mathrm{m}}_{\mathbb{D}}(P_m)$ belongs to the algebra of level-$4$ cyclotomic multiple polylogarithm values, and we evaluate the first nontrivial case $m=2$ explicitly in terms of ${\mathrm{Li}}_4(1/2)$, $\zeta (3)$, Catalan's constant, $\pi$, and $\log 2$. We also derive an explicit three-term asymptotic expansion for ${\mathrm{m}}_{\mathbb{D}}(P_m)$ as $m\to \infty$.

28 pages; substantially revised and expanded from v1. Added two main theorems on cyclotomic special values and an explicit formula for m=2