Linear Mahler measures and double \(L\)-values of modular forms
type: Item
instance of: scholarly article
title: Linear Mahler measures and double \(L\)-values of modular forms
zbMATH Open document ID: 1298.11100
author: Evgeny Shinder
author: Masha Vlasenko
published in: Journal of Number Theory
publication date: 2014-08-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1206.1454
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 11R06
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F67
zbMATH DE Number: 6335916
zbMATH Keywords: Mahler measure
zbMATH Keywords: special values of \(L\)-functions
zbMATH Keywords: modular forms
MaRDI profile type: Publication
OpenAlex ID: W2042564957
arXiv ID: 1206.1454
cites work: Q4013269
cites work: Q4252760
cites work: ESTIMATES FOR MAHLER’S MEASURE OF A LINEAR FORM
cites work: On Measures of Polynomials in Several Variables: Corrigendum
cites work: On the Picard-Fuchs equation and the formal Brauer group of certain elliptic \(K3\)-surfaces
cites work: Root lattices and pencils of varieties
cites work: Apéry limits and special values of \(L\)-functions
cites work: Elliptic Modular Forms and Their Applications
cites work: Irrationality of values of the Riemann zeta function
DOI: 10.1016/J.JNT.2014.02.005
review text: The equality NEWLINE\[NEWLINEm(P)={1\over2\pi i)^n}\int_{|z_1|=\cdots=|z_n|=1}{\log|P(x_1,\dots,x_n)|\over x_1\cdots x_n}dx_1\cdots x_nNEWLINE\]NEWLINE defines the logarithm of Mahler measure of a polynomial \(P(x_1,\dots,x_n)\). It has been shown by \textit{C. J. Smyth} [Bull. Aust. Math. Soc. 23, 49--63 (1981; Zbl 0442.10034)] that NEWLINE\[NEWLINEm(1+x_1+x_2)={3\sqrt3\over4\pi}L(\chi_{3},2),\tag{1}NEWLINE\]NEWLINE and in Appendix 1 of \textit{D. W. Boyd}'s paper [Can. Math. Bull. 24, 453--469 (1981; Zbl 0474.12005)]NEWLINE\[NEWLINEm(1+x_1+x_2+x_3) = {7\over2\pi^2}\zeta(3).\tag{2}NEWLINE\]NEWLINENEWLINENEWLINENEWLINEThe author obtains a rather complicated formula expressing \(m(1+x_1+x_2+x_3+x_4)\) by integrals of modular forms.
description: scientific article; zbMATH DE number 6335916