Can polylogarithms at algebraic points be linearly independent?

Abstract

Let $r,m$ be positive integers. Let $0\le x<1$ be a rational number. Let ${\Phi }_s(x,z)$ be the $s$-th Lerch function ${\sum }_{k=0}^{\infty }\frac{z^{k+1}}{(k+x+1{)}^s}$ 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 $0<|{\alpha }_j|<1$ $(1\le j\le m)$. In this article, we state a linear independence criterion over algebraic number fields of all the $rm+1$ numbers $:$ ${\Phi }_1(x,{\alpha }_1),{\Phi }_2(x,{\alpha }_1),\dots ,{\Phi }_r(x,{\alpha }_1),{\Phi }_1(x,{\alpha }_2),{\Phi }_2(x,{\alpha }_2),\dots ,{\Phi }_r(x,{\alpha }_2),\dots ,{\Phi }_1(x,{\alpha }_m),{\Phi }_2(x,{\alpha }_m),\dots ,{\Phi }_r(x,{\alpha }_m)$ and $1$. This is the first result that gives a sufficient condition for the linear independence of values of the $r$ Lerch functions ${\Phi }_1(x,z),{\Phi }_2(x,z),\dots ,{\Phi }_r(x,z)$ at $m$ distinct algebraic points without any assumption for $r$ and $m$, even for the case $x=0$, the polylogarithms. We give an outline of our proof and explain basic idea.

Corrected typos

References in corpus (1)

• Many odd zeta values are irrational

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