Analytic renormalization of multiple zeta functions. Geometry and combinatorics of the generalized Euler reflection formula for MZV

TL;DR

This paper introduces a geometric and combinatorial approach to renormalizing multiple zeta functions, revealing new insights into their values, connections to the inverse Gamma function, and links to the generalized Euler reflection formula.

Contribution

It demonstrates that renormalization can be achieved through calculus methods, providing a geometric interpretation and explicit formulas for multiple zeta values at non-convergent arguments.

Findings

Renormalized multiple zeta values relate to the inverse Gamma function's asymptotic expansion.

The combinatorics match the generalized Euler reflection formula.

A new function interpolates the Riemann zeta function and Euler constant.

Abstract

The renormalization of MZV was until now carried out by algebraic means. We show that renormalization in general, of the multiple zeta functions in particular, is more than mere convention. We show that simple calculus methods allow us to compute the renormalized values of multiple zeta functions in any dimension for arguments of the form (1,...,1), where the series do not converge. These values happen to be the coefficients of the asymptotic expansion of the inverse Gamma function. We focus on the geometric interpretation of these values, and on the combinatorics their closed form encodes, which happen to match the combinatorics of the generalized Euler reflection formula discovered by Michael E. Hoffman, which in turn is a kind of analogue of the Cayley-Hamilton theorem for matrices. By means of one single limit formula, we define a function on the positive open half-line which takes exactly the values of the Riemann zeta function, with the additional advantage that it equals the Euler constant when the argument is 1.