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

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

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