Probability Laws Concerning Zeta Integrals

TL;DR

This paper provides a probabilistic interpretation of Dedekind zeta functions for specific quadratic fields, demonstrating the positivity of their first two Li coefficients, extending previous results on the Riemann zeta function.

Contribution

It introduces a novel probabilistic framework for Dedekind zeta functions of quadratic fields and proves positivity of initial Li coefficients for these cases.

Findings

Probabilistic interpretation of Dedekind zeta functions for bQ(4) and bQ(2)

Positivity of the first two Li coefficients for these zeta functions

Extension of previous results from the Riemann zeta function case

Abstract

We give a probabilistic interpretation of the Dedekind zeta functions of $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-2})$ using zeta integrals and use this to show that the first two Li coefficients of these zeta functions are positive. This extends a result of Biane, Pitman, and Yor (2001) which considered the case of the Riemann zeta function.