# A penalised model reproducing the mod-Poisson fluctuations in the Sathe-Selberg theorem

## TL;DR

This paper introduces a penalized probabilistic model that accurately reproduces the mod-Poisson fluctuations observed in the distribution of divisors of random integers, offering a new perspective on classical number theory results.

## Contribution

It presents a novel penalized model that converges in the mod-Poisson sense, providing an alternative to existing hybrid product models for divisor distributions.

## Key findings

- Model reproduces mod-Poisson fluctuations accurately
- Offers an alternative probabilistic perspective
- Contributes to understanding divisor distribution in number theory

## Abstract

We construct a probabilistic model for the number of divisors of a random uniform integer that converges in the mod-Poisson sense to the same limiting function as its original counterpart, the one arising in the Sathe-Selberg theorem. This construction involves a conditioning and gives an alternative perspective to the usual paradigm of "hybrid product" models developed by Gonek, Hughes and Keating in the case of the Riemann Zeta function.