Probabilistic renormalization and analytic continuation

TL;DR

This paper develops a probabilistic framework for renormalizing series, linking it with analytic continuation, and applies it to the Riemann zeta function, providing new insights into series regularization.

Contribution

It introduces a novel probabilistic renormalization method for series, especially Dirichlet series, and demonstrates its compatibility with analytic continuation.

Findings

Probabilistic renormalization encodes series values as expectations of random variables.

A class of weakly renormalizable Dirichlet series is identified and analyzed.

The Riemann zeta function's renormalized value aligns with its classical value for s ≠ 1.

Abstract

We introduce a theory of probabilistic renormalization for series, the renormalized values being encoded in the expectation of a certain random variable on the set of natural numbers. We identify a large class of weakly renormalizable series of Dirichlet type, whose analysis depends on the properties of a (infinite order) difference operator that we call Bernoulli operator. For the series in this class, we show that the probabilistic renormalization is compatible with analytic continuation. The general zeta series for $s\neq 1$ is found to be strongly renormalizable and its renormalized value is given by the Riemann zeta function.