Polylogarithms in Arithmetic and Geometry

Abstract

The classical polylogarithms were invented in the correspondence of Leibniz with Bernoulli in 1696 [Lei]. They are defined by the series {\$}{\$}L{\{}i{\_}n{\}}(z) = {\backslash}sum{\backslash}limits{\_}{\{}k = 1{\}}^{\backslash}infty {\{}{\backslash}frac{\{}{\{}{\{}z^k{\}}{\}}{\}}{\{}{\{}{\{}k^n{\}}{\}}{\}}{\}} {\backslash}left| z {\backslash}right| < 1{\$}{\$}Lin(z)=∑k=1∞zkkn|z|<1and continued analytically to a covering of ℂP1{\backslash}{\{}0,1,∞{\}}:{\$}{\$}L{\{}i{\_}n{\}}(z): = {\backslash}int {\{}{\_}0^zL{\{}i{\_}{\{}n - 1{\}}{\}}(t){\backslash}frac{\{}{\{}dt{\}}{\}}{\{}t{\}},L{\{}i{\_}1{\}}(z) = - {\backslash}log (1 - z){\}}{\$}{\$}Lin(z):=∫0zLin−1(t)dtt,Li1(z)=−log(1−z).