Three-variable Mahler measures and special values of \(L\)-functions of modular forms
Mahler's measures of the two families of Laurent polynomials \[ P_{k}(x,y,z)=(x+\frac{1}{x})(y+\frac{1}{y})(z+\frac{1}{z})+\sqrt{k}, \] and \[ Q_{k}(x,y,z)=x^{4}+y^{4}+z^{4}+1+k^{1/4}xyz, \] where \(k\in \mathbb{C},\) were considered by \textit{M. D. Rogers} [Ramanujan J. 18, No. 3, 327--340 (2009; Zbl 1226.11113)], and then by \textit{D. Samart} [Ramanujan J. 32, No. 2, 245--268 (2013; Zbl 1356.11029); Can. J. Math. 67, No. 2, 424--449 (2015; Zbl 1344.11070)], who proposed identities between some of these measures and special values of \(L\)-functions of certain modular forms. Seven of such conjectural identities, namely those related to measures of \[ P_{-64},\ P_{-512},\ P_{280+198\sqrt{2}},\ P_{280-198\sqrt{2}},\ Q_{-1024},\ Q_{-12288}\text{ and }Q_{-82944}, \] are proven in the paper under review. Parts of the proof of this theorem use arguments similar to those employed in [\textit{H. Zheng} et al., Electron Res. Arch. 28, No. 1, 103--125 (2020; Zbl 1439.11276)] to show analogous results.
scientific article; zbMATH DE number 7345304
scientific article