A New Probabilistic Representation of the Alternating Zeta Function and a New Selberg-like Integral Evaluation

TL;DR

This paper introduces new representations of the alternating Zeta function using determinants and probabilistic expectations, enabling the evaluation of a Selberg-like integral with a generalized Vandermonde determinant.

Contribution

It provides novel determinant-based and probabilistic representations of the alternating Zeta function, extending to a generalized Selberg integral evaluation.

Findings

Representation of the alternating Zeta function as a limit of determinants

Expression of determinants as expectations of functionals of Dixon-Anderson distributed vectors

Evaluation of a generalized Selberg-type integral

Abstract

In this paper, we present two new representations of the alternating Zeta function. We show that for any s $\in$ C this function can be computed as a limit of a series of determinant. We then express these determinants as the expectation of a functional of a random vector with Dixon-Anderson density. The generalization of this representation to more general alternating series allows us to evaluate a Selberg-type integral with a generalized Vandermonde determinant.