The Mahler measure of \((x+1/x)(y+1/y)(z+1/z)+\sqrt{k}\)
The (logarithmic) Mahler measure $m(P)$ of a non-zero Laurent polynomial $P\in \mathbb{C}[X_{1}^{\pm 1},\dots,X_{n}^{\pm 1}]$ is defined by \[ m(P)=\int_{0}^{1}\cdots \int_{0}^{1}\log \left\vert P(\exp (i2\pi t_{1}),\dots,\exp (i2\pi t_{n})\right\vert dt_{1}\cdots dt_{n}. \] It is well known that there are non-trivial relations between Mahler measures of certain types of Laurent polynomials and special values of $L$-functions. In this context, the family of reciprocal polynomials \[ P_{k}(x,y,z)=(x+1/x)(y+1/y)(z+1/z)+\sqrt{k}, \] where $k$ is a constant, has been considered by \textit{M. D. Rogers} [Ramanujan J. 18, No. 3, 327--340 (2009; Zbl 1226.11113)], and then by \textit{D. Samart} [Can. J. Math. 67, No. 2, 424--449 (2015; Zbl 1344.11070)] who proposed identities between some measures $m(P_{k})$ and values of $L$-functions of certain modular forms; six of such conjectural identities are proved in the paper under review.
scientific article; zbMATH DE number 7189501
scientific article