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Approximation of Smooth Numbers for Harmonic Samples A Stein method Approach

Arturo Jaramillo, Xiaochuan Yang

TL;DR

This work develops a probabilistic framework for understanding the content of $m$-smooth numbers up to $n$ by introducing $ Psi_H[m,n]$, the harmonic-measure analogue of the classical count $ Psi(n,m)$. The authors model prime multiplicities via independent geometric random variables and prove a sharp Dickman-type approximation using Stein's method for the Dickman distribution, with key regularity inputs from Bhattacharjee and Schulte. The main result provides a uniform, explicit sup-norm approximation: $ Psi_H(n,m) = rac{n}{6(n,m)} I[ ho](6(n,m)) + O(1/ log n)$ for $16\le m\nle n$, where $6(n,m)= rac{ log n}{ log m}$ and $ I[ ho](x)= int_0^x ho(s)\,ds$, connecting to the Dickman function $ ho$. The analysis relies on a two-step program: (i) a stochastic representation of $H_n$ via $oldsymbol{\xi_p}$ and convergence to the Dickman law, and (ii) a sup-norm bound derived from the Dickman Stein equation and Mertens formulas; these results situate a probabilistic route alongside classical de Bruijn-type asymptotics and offer a framework potentially informative for questions related to Hildebrand’s criterion and the Riemann hypothesis.

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.

Approximation of Smooth Numbers for Harmonic Samples A Stein method Approach

TL;DR

This work develops a probabilistic framework for understanding the content of -smooth numbers up to by introducing , the harmonic-measure analogue of the classical count . The authors model prime multiplicities via independent geometric random variables and prove a sharp Dickman-type approximation using Stein's method for the Dickman distribution, with key regularity inputs from Bhattacharjee and Schulte. The main result provides a uniform, explicit sup-norm approximation: for , where and , connecting to the Dickman function . The analysis relies on a two-step program: (i) a stochastic representation of via and convergence to the Dickman law, and (ii) a sup-norm bound derived from the Dickman Stein equation and Mertens formulas; these results situate a probabilistic route alongside classical de Bruijn-type asymptotics and offer a framework potentially informative for questions related to Hildebrand’s criterion and the Riemann hypothesis.

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.
Paper Structure (14 sections, 10 theorems, 104 equations)

This paper contains 14 sections, 10 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Overview on the asymptotics for the size of $m$-smooth numbers
  3. Some questions for future research
  4. An overview on the description of $\Psi_{H}$
  5. Some known approximation results for $\Psi(n,m)$
  6. The function $\Psi$ and its relation to the Dickman function
  7. Main results
  8. Preliminaries
  9. Dickman distributional approximations
  10. Mertens' formulas
  11. The geometric distribution and the multiplicities of Harmonic samples
  12. Proof of Theorem \ref{['thm:mainone']}
  13. Proof of Lemma \ref{['lem:simplificationtwo']}
  14. Proof of Lemma \ref{['lem:kolmmain']}

Key Result

Theorem 2.1

For a fixed value of $\varepsilon>0$, define the set Then, there exists a constant $C_{\varepsilon}$ only depending on $\varepsilon$, such that where Otherwise said, it holds that

Theorems & Definitions (14)

  • Theorem 2.1: de Bruijn, 1951
  • Theorem 2.2: Hildebrand, 1986
  • Remark 2.1
  • Theorem 2.3: Hildebrand, 1984
  • Remark 2.2
  • Theorem 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Proposition 4.1
  • Lemma 4.3
  • ...and 4 more