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Probabilistic interpretation of the Selberg--Delange Method in analytic number theory

Maximilian Janisch

TL;DR

This work connects the analytic Selberg–Delange method with probabilistic mod-Poisson convergence to study the number of distinct prime factors ω(N_{x,α}) of a random integer weighted by a broad class of multiplicative functions α. Under the admissible++ framework, the authors prove that ω(N_{x,α}) converges mod-Poisson on the entire complex plane with limiting function ψ(z)=λ0(α_{e^z})/λ0(α) and t_x=ρ ln ln x, with a rate O(1/ln x); they also extend to general additive functions g via α_y constructions. Consequences include a Central Limit Theorem and precise large deviations for ω(N_{x,α}), providing refined probabilistic descriptions beyond Erdős–Kac-type results. The paper further showcases several admissible++ examples, demonstrating the versatility of the approach for multiplicative weights and highlighting connections to recent CLTs in the literature.

Abstract

In analytic number theory, the Selberg--Delange Method provides an asymptotic formula for the partial sums of a complex function $f$ whose Dirichlet series has the form of a product of a well-behaved analytic function and a complex power of the Riemann zeta function. In probability theory, mod-Poisson convergence is a refinement of convergence in distribution toward a normal distribution. This stronger form of convergence not only implies a Central Limit Theorem but also offers finer control over the distribution of the variables, such as precise estimates for large deviations. In this paper, we show that results in analytic number theory derived using the Selberg--Delange Method lead to mod-Poisson convergence as $x \to \infty$ for the number of distinct prime factors of a randomly chosen integer between $1$ and $x$, where the integer is distributed according to a broad class of multiplicative functions. As a Corollary, we recover a part of a recent result by Elboim and Gorodetsky under different, though related, conditions: A Central Limit Theorem for the number of distinct prime factors of such random integers.

Probabilistic interpretation of the Selberg--Delange Method in analytic number theory

TL;DR

This work connects the analytic Selberg–Delange method with probabilistic mod-Poisson convergence to study the number of distinct prime factors ω(N_{x,α}) of a random integer weighted by a broad class of multiplicative functions α. Under the admissible++ framework, the authors prove that ω(N_{x,α}) converges mod-Poisson on the entire complex plane with limiting function ψ(z)=λ0(α_{e^z})/λ0(α) and t_x=ρ ln ln x, with a rate O(1/ln x); they also extend to general additive functions g via α_y constructions. Consequences include a Central Limit Theorem and precise large deviations for ω(N_{x,α}), providing refined probabilistic descriptions beyond Erdős–Kac-type results. The paper further showcases several admissible++ examples, demonstrating the versatility of the approach for multiplicative weights and highlighting connections to recent CLTs in the literature.

Abstract

In analytic number theory, the Selberg--Delange Method provides an asymptotic formula for the partial sums of a complex function whose Dirichlet series has the form of a product of a well-behaved analytic function and a complex power of the Riemann zeta function. In probability theory, mod-Poisson convergence is a refinement of convergence in distribution toward a normal distribution. This stronger form of convergence not only implies a Central Limit Theorem but also offers finer control over the distribution of the variables, such as precise estimates for large deviations. In this paper, we show that results in analytic number theory derived using the Selberg--Delange Method lead to mod-Poisson convergence as for the number of distinct prime factors of a randomly chosen integer between and , where the integer is distributed according to a broad class of multiplicative functions. As a Corollary, we recover a part of a recent result by Elboim and Gorodetsky under different, though related, conditions: A Central Limit Theorem for the number of distinct prime factors of such random integers.

Paper Structure

This paper contains 14 sections, 11 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Admissible functions
  4. Recall of mod-phi convergence
  5. Statement of main result and its proof
  6. Generalization to additive functions other than omega
  7. Consequences
  8. Central Limit Theorem
  9. Precise large deviations
  10. Examples of admissible++ functions
  11. alpha(k)=theta pow omega(k) for a theta>0
  12. alpha(ps)=Bs for a B in (0,2)
  13. alpha(p)=a+O(p pow - epsilon k) for some epsilon>0
  14. Proof of Proposition

Key Result

theorem 1

Let $\alpha:\mathbb{N}\to[0,\infty)$ be a multiplicative function that is not identically equal to $0$. If $\alpha$ is admissible++, a term that will be defined later in the text, see Definition and Definition , and $(N_{x,\alpha})_{x\in\mathbb{N}}$ is a family of random variables such that $\mathbb{P}(N_{x,\alpha}\in\set{1,\dots, x})=1$ and such that $\mathbb{P}(N_{x,\alpha}=k)$ is proportional

Theorems & Definitions (32)

  • definition 1: Multiplicative functions
  • theorem 1: Main result, shortened version of Theorem
  • definition 2: Admissibility of functions
  • Example 2: Examples of admissible functions
  • definition 3: Admissibility++
  • remark 1: Discussion of admissibility++
  • remark 2: The condition that $\delta=1$
  • remark 3: Is the notion of admissibility++ necessary?
  • remark 4: Uniformity of the convergence in \ref{['eq:sum-of-squares']}
  • proposition 1: Classes of admissible functions
  • ...and 22 more