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

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.