Large deviations for number of irreducible divisors of the Dirichlet series distribution

TL;DR

This paper develops precise large deviation estimates for the number of irreducible divisors in distributions related to Dirichlet series, using mod-Poisson convergence, with applications in number theory and finite fields.

Contribution

It introduces a general method for large deviation estimates in number-theoretic distributions based on Dirichlet series, utilizing mod-Poisson convergence.

Findings

Derived explicit large deviation estimates for irreducible divisors

Applied the method to number theory, Dedekind domains, and finite fields

Demonstrated the effectiveness of mod-Poisson convergence in this context

Abstract

In this paper we produce precise large deviation estimates through the lens of mod-Poisson convergence. We apply a general result to various examples from number theory, Dedekind domains and polynomials over finite fields when an element is selected using a distribution based on a Dirichlet series.