#### 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 $0\le x<1$ be a rational number. We denote by ${\Phi }_{s}\left(x,z\right)$ the $s$-th Lerch function

$\sum _{k=0}^{\infty }\frac{{z}^{k+1}}{{\left(k+x+1\right)}^{s}},$

with $s=1,2,\dots ,r$. When $x=0$, this is the polylogarithmic function. Let ${\alpha }_{1},\dots ,{\alpha }_{m}$ be pairwise distinct algebraic numbers with $0<|{\alpha }_{j}|<1$ ($1\le j\le m$). We state a linear independence criterion over algebraic number fields of all the $rm+1$ numbers: ${\Phi }_{1}\left(x,{\alpha }_{1}\right)$, ${\Phi }_{2}\left(x,{\alpha }_{1}\right)$, $\dots \phantom{\rule{0.3em}{0ex}}$, ${\Phi }_{r}\left(x,{\alpha }_{1}\right)$, ${\Phi }_{1}\left(x,{\alpha }_{2}\right)$, ${\Phi }_{2}\left(x,{\alpha }_{2}\right)$, $\dots \phantom{\rule{0.3em}{0ex}}$, ${\Phi }_{r}\left(x,{\alpha }_{2}\right)$, $\dots \phantom{\rule{0.3em}{0ex}}$, ${\Phi }_{1}\left(x,{\alpha }_{m}\right)$, ${\Phi }_{2}\left(x,{\alpha }_{m}\right)$, $\dots \phantom{\rule{0.3em}{0ex}}$, ${\Phi }_{r}\left(x,{\alpha }_{m}\right)$ and $1$. We obtain an explicit sufficient condition for the linear independence of values of the $r$ Lerch functions ${\Phi }_{1}\left(x,z\right)$, $\dots \phantom{\rule{0.3em}{0ex}}$, ${\Phi }_{r}\left(x,z\right)$ 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
##### 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