Vol. 9, No. 4, 2020

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Can polylogarithms at algebraic points be linearly independent?

Sinnou David, Noriko Hirata-Kohno and Makoto Kawashima

Vol. 9 (2020), No. 4, 389–406
Abstract

Let r,m be positive integers. Let 0 x < 1 be a rational number. We denote by Φs(x,z) the s-th Lerch function

k=0 zk+1 (k + x + 1)s,

with s = 1,2,,r. When x = 0, this is the polylogarithmic function. Let α1,,αm be pairwise distinct algebraic numbers with 0 < |αj| < 1 (1 j m). We state a linear independence criterion over algebraic number fields of all the rm + 1 numbers: Φ1(x,α1), Φ2(x,α1), , Φr(x,α1), Φ1(x,α2), Φ2(x,α2), , Φr(x,α2), , Φ1(x,αm), Φ2(x,αm), , Φr(x,αm) and 1. We obtain an explicit sufficient condition for the linear independence of values of the r Lerch functions Φ1(x,z), , Φr(x,z) at m distinct points in an algebraic number field of arbitrary finite degree without any assumptions on r and m. When x = 0, our result implies the linear independence of polylogarithms of distinct algebraic numbers of arbitrary degree, subject to a metric condition. We give an outline of our proof together with concrete examples of linearly independent polylogarithms.

Dedicated to the memory of Professor Naum Ilyitch Feldman

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Keywords
Lerch function, polylogarithms, linear independence, irrationality, Padé approximation
Mathematical Subject Classification 2010
Primary: 11G55, 11J72, 11J82, 11J86, 11M35
Secondary: 11D75, 11D88
Milestones
Received: 10 December 2019
Revised: 1 May 2020
Accepted: 15 May 2020
Published: 5 November 2020
Authors
Sinnou David
Institut de Mathématiques de Jussieu-Paris Rive Gauche
CNRS UMR 7586
Sorbonne Université
Paris
France
CNRS UMI 2000 Relax
Chennai Mathematical Institute
Kelambakkam
India
Noriko Hirata-Kohno
Department of Mathematics
College of Science and Technology
Nihon University
Tokyo
Japan
Makoto Kawashima
Department of Liberal Arts and Basic Sciences
College of Industrial Engineering
Nihon University
Chiba
Japan