Distribution Laws of Smooth Divisors

TL;DR

This paper studies the distribution of smooth divisors of integers, revealing that on average these divisors follow a specific probability law, extending classical divisor distribution results.

Contribution

It introduces a new distribution law for smooth divisors, generalizing classical divisor distribution results to smooth divisors.

Findings

Smooth divisors follow a distinct probability distribution.

The distribution extends classical divisor laws to smooth divisors.

Provides a theoretical framework for understanding smooth divisor distribution.

Abstract

A classical result due to Deshouillers, Dress and Tenenbaum asserts that on average the distribution of the divisors of the integers follows the arcsine law. In this paper, we investigate the distribution of smooth divisors of the integers, that is, those divisors which are free of large prime factors. We show that on average these divisors are distributed according to a probability law that we will describe.