Nested sums of symbols and renormalised multiple zeta functions

TL;DR

This paper introduces a framework for defining and renormalizing nested sums of symbols on the real line, establishing their relations to multiple zeta functions and proving rationality results at nonpositive integers.

Contribution

It develops a novel method for renormalizing nested sums of symbols using holomorphic regularisation and Birkhoff factorisation, extending to multiple zeta functions and their properties.

Findings

Renormalised nested sums satisfy stuffle relations at all arguments.

Multiple zeta values at nonpositive integers are rational.

Framework includes Hurwitz multiple zeta functions and higher-dimensional analogs.

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.