Nested sums of symbols and renormalised multiple zeta functions

2007/02/06 by Dominique Manchon, Manchon, Dominique, Sylvie Paycha +1 · 1 citation
Mathematics · #11M99 #35S05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Functional Analysis (math.FA) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0702135

openalex publication_date 2007/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define discrete nested sums over integer points for symbols on the real line, which obey stuffle relations whenever they converge. They relate to Chen integrals of symbols via the Euler-MacLaurin formula. Using a suitable holomorphic regularisation followed by a Birkhoff factorisation, we define renormalised nested sums of symbols which also satisfy stuffle relations. For appropriate symbols they give rise to renormalised multiple zeta functions which satisfy stuffle relations at all arguments. The Hurwitz multiple zeta functions fit into the framework as well. We show the rationality of multiple zeta values at nonpositive integer arguments, and a higher-dimensional analog is also investigated.

Citations

• Multiple polylogarithms and mixed Tate motives

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