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

TL;DR

The paper develops a unified mod-$\text{Poisson}$ framework for Dirichlet-series–based distributions on suitable semigroups and proves precise large deviation and Berry–Esseen-type results for counts of irreducible divisors, both without and with multiplicity. It identifies explicit mean-like parameters $t_s$ and residue functions $\psi$ and shows mod-$\text{Poisson}$ convergence under a set of verifiable assumptions ${\bf A1}$–${\bf A5}$ (and ${\bf A0}$, ${\bf A6}$), yielding exact asymptotics for $\omega(X_s)$ and $\Omega(X_s)$ as $s\downarrow\kappa$. The results are illustrated through numeric examples with classical zeta-type distributions (Riemann, Dedekind, Euler, divisor-counting) and extended to non-numeric settings such as random polynomials over finite fields and random ideals in Dedekind domains. Overall, the work provides sharp, distribution-accurate large-deviation formulas and robust probabilistic bounds that unify several counting problems across number theory and algebraic structures. These findings have potential applications in probabilistic number theory, random algebraic structures, and computational sampling schemes driven by Dirichlet-series distributions.

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.

Theorems & Definitions (41)

• Definition 1.1
• Definition 2.1
• Lemma 2.1
• Theorem 2.1
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• Theorem 3.3