2000/06/07 by Amnon Besser, Besser, Amnon
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #K-Theory and Homology (math.KT) #Number Theory (math.NT) #math.KT #math.NT
paper · pdf · doi:10.48550/arxiv.math/0006051
7 pages, latex2e with amsart class
arxiv created 2000/06/07 · openalex publication_date 2000/06/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The finite n-th polylogarithm lin(z) in Z/p[z] is defined as the sum on k from 1 to p-1 of zk/kn. We state and prove the following theorem. Let Lik:Cp to Cp be the p-adic polylogarithms defined by Coleman. Then a certain linear combination Fn of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p1-n DFn(zp) reduces modulo p>n+1 to lin-1(z) where D is the Cathelineau operator z(1-z) d/dz. A slightly modified version of this theorem was conjectured by Kontsevich. This theorem is used by Elbaz-Vincent and Gangl to deduce functional equations of finite polylogarithms from those of complex polylogarithms.