An equivalence between desingularized and renormalized values of multiple zeta functions at negative integers

TL;DR

This paper establishes a precise relationship between desingularized and renormalized multiple zeta values at negative integers, providing a unified understanding and explicit formulas involving Bernoulli numbers.

Contribution

It reveals an explicit equivalence between two different approaches to defining multiple zeta values at negative integers, connecting desingularization and renormalization methods.

Findings

Proves an explicit interrelationship between desingularized and renormalized values.

Derives a formula expressing renormalized values in terms of Bernoulli numbers.

Provides a unified framework for understanding special values of multiple zeta functions at negative integers.

Abstract

It is known that the special values of multiple zeta functions at non-positive arguments are indeterminate in most cases due to the occurrences of infinitely many singularities. In order to give a suitable rigorous meaning of the special values there, Furusho, Komori, Matsumoto and Tsumura introduced the desingularized values by the desingularization method to resolve all singularities. While, Ebrahimi-Fard, Manchon and Singer introduced the renormalized values to keep the "shuffle" relation by the renormalization procedure \`a la Connes and Kreimer. In this paper, we reveal an equivalence, that is, an explicit interrelationship between these two values. As a corollary, we also obtain an explicit formula to describe renormalized values in terms of Bernoulli numbers.