Shuffle-type product formulae of desingularized values of multiple zeta-functions

TL;DR

This paper establishes shuffle-type product formulae for desingularized values of multiple zeta-functions, providing a rigorous framework to handle their singularities and indeterminate values at non-positive integers.

Contribution

It introduces shuffle-type product formulae for desingularized multiple zeta-values, bridging the gap between desingularization and algebraic product structures.

Findings

Proves shuffle-type product formulae for desingularized values.

Shows desingularized values satisfy algebraic relations similar to renormalized values.

Provides a rigorous foundation for evaluating multiple zeta-functions at singular points.

Abstract

It is known that there are infinitely many singularities of multiple zeta functions and the special values at non-positive integer points are indeterminate. In order to give a suitable rigorous meaning of the special values there, Furusho, Komori, Matsumoto and Tsumura introduced desingularized values by using their desingularization method to resolve all singularities. On the other hand, Ebrahimi-Fard, Manchon and Singer introduced renormalized values by the renormalization method \`{a} la Connes and Kreimer and they showed that the values fulfill the shuffle-type product formula. In this paper, we show the shuffle-type product formulae for desingularized values.