2006/11/08 by Charles L. Epstein, Epstein, Charles L., Jack Morava +1
Mathematics · #11G55 #46F20 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #math.CA #math.NT #msc:11G55 #msc:46F20
paper · pdf · doi:10.48550/arxiv.math/0611240
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arxiv created 2006/11/08 · openalex publication_date 2006/11/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the function Lis(ex) extends to an entire function of the complex variable s, taking values in tempered distributions in x on the whole real line. That the classical polylogarithm extends similarly, as an entire function taking values in distributions on compactly-supported functions on the positive real axis, is a corollary. We then identify the singularities of Lis(ex) in terms of distribution powers of x; this leads to a simple proof of the smoothness of the `modified' polylogarithm of Bloch, Ramakrishnan, Wigner, Wojtkowiak, Zagier, and others.