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2020 Can polylogarithms at algebraic points be linearly independent?
Sinnou David, Noriko Hirata-Kohno, Makoto Kawashima
Mosc. J. Comb. Number Theory 9(4): 389-406 (2020). DOI: 10.2140/moscow.2020.9.389

Abstract

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

k = 0 z k + 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 (1jm). 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.

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Sinnou David. Noriko Hirata-Kohno. Makoto Kawashima. "Can polylogarithms at algebraic points be linearly independent?." Mosc. J. Comb. Number Theory 9 (4) 389 - 406, 2020. https://doi.org/10.2140/moscow.2020.9.389

Information

Received: 10 December 2019; Revised: 1 May 2020; Accepted: 15 May 2020; Published: 2020
First available in Project Euclid: 12 November 2020

zbMATH: 07272357
MathSciNet: MR4170704
Digital Object Identifier: 10.2140/moscow.2020.9.389

Subjects:
Primary: 11G55, 11J72, 11J82, 11J86, 11M35
Secondary: 11D75, 11D88

Rights: Copyright © 2020 Mathematical Sciences Publishers

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