The Mahler measure of \((x+1/x)(y+1/y)(z+1/z)+\sqrt{k}\)
type: Item
instance of: scholarly article
title: The Mahler measure of \((x+1/x)(y+1/y)(z+1/z)+\sqrt{k}\)
zbMATH Open document ID: 1439.11276
author: Huimin Zheng
author: Hourong Qin
author: Xuejun Guo
published in: Electronic Research Archive
publication date: 2020-04-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 33C20
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 7189501
zbMATH Keywords: Mahler measure
zbMATH Keywords: modular forms
zbMATH Keywords: Dirichlet series
zbMATH Keywords: \(L\)-functions
zbMATH Keywords: Hecke characters
reviewed by: Toufik Zaïmi
describes a project that uses: SageMath
MaRDI profile type: Publication
full work available at URL: era.2020007
OpenAlex ID: W3002589010
cites work: Q3438252
cites work: The Mahler measure for \(K3\) hypersurfaces
cites work: Weber's class invariants
cites work: Mahler's Measure and Special Values of L-functions
cites work: Deligne periods of mixed motives, $K$-theory and the entropy of certain $\mathbb {Z}^n$-actions
cites work: Three-variable Mahler measures and special values of \(L\)-functions of modular forms
cites work: Functional equations for Mahler measures of genus-one
cites work: Q4449110
cites work: New \(_{5}F_{4}\) hypergeometric transformations, three-variable Mahler measures, and formulas for \(1/ \pi \)
cites work: FromL-series of elliptic curves to Mahler measures
cites work: On the Mahler measure of \(1+X+1/X+Y +1/Y\)
cites work: Q4252760
cites work: Three-variable Mahler measures and special values of modular and Dirichlet \(L\)-series
cites work: Mahler Measures as Linear Combinations ofL–values of Multiple Modular Forms
cites work: On Measures of Polynomials in Several Variables: Corrigendum
cites work: On the singular values of Weber modular functions
DOI: 10.3934/ERA.2020007
description: scientific article; zbMATH DE number 7189501