Log-sine evaluations of Mahler measures (Q2898883)

The authors obtain some new formulas for the multiple Mahler measure \(\mu(P_1,\dots,P_k)\) and the higher Mahler measure \(\mu_k(P)=\mu(P,\dots,P)\), where the argument \(P\) is taken \(k\) times, in terms of polylogarithmic terms. They prove, for instance, that NEWLINE\[NEWLINE\mu_2(1+x+y)=\frac{\pi^2}{4} - \frac{3}{\pi} \int_{0}^{\pi/3} \log^2(2 \cos(\theta/2))\, d\thetaNEWLINE\]NEWLINE which disproves an earlier incorrect equality of \textit{N. Kurokawa, M. Lalin} and \textit{H. Ochiai} [Acta Arith. 135, No. 3, 269--297 (2008; Zbl 1211.11116)]. They also find the formulas (with few polylogarithmic terms) for \(\mu_k(1+x,\dots,1+x,1+x+y)\), where the argument \(1+x\) appears \(k-1\) times and \(k=1, \dots, 6\), e.g., \(\mu_2(1+x,1+x+y)=\pi^2/54\).