Approximation of Smooth Numbers for Harmonic Samples A Stein method Approach

TL;DR

This paper develops a Stein method-based approximation for the distribution of smooth numbers, using probabilistic representations and recent regularity estimates, bridging number theory and probability.

Contribution

It introduces a novel Stein method approach for approximating smooth number distributions, leveraging probabilistic models and recent analytical results.

Findings

Provides a de Bruijn type approximation for smooth numbers.

Utilizes weighted geometric random variables for stochastic representation.

Relies mainly on probability theory with minimal number theory estimates.

Abstract

We present a de Bruijn type approximation for quantifying the content of m smooth numbers, derived from samples obtained through a probability measure over the set of integers less than or equal to n, with point mass function at k inversely proportional to k. Our analysis is based on a stochastic representation of the measure of interest, utilizing weighted independent geometric random variables. This representation is analyzed through the lens of Stein method for the Dickman distribution. A pivotal element of our arguments relies on precise estimations concerning the regularity properties of the solution to the Dickman Stein equation for heaviside functions, recently developed by Bhattacharjee and Schulte. Remarkably, our arguments remain mostly in the realm of probability theory, with Mertens first and third theorems standing as the only number theory estimations required.