On the Distribution of Integers with Restricted Prime Factors I

TL;DR

This paper derives an asymptotic formula for counting integers with prime factors in specified sets, revealing deviations from probabilistic models and extending results on integers with prime factors in arithmetic progressions.

Contribution

It provides a new asymptotic formula for the distribution of integers with restricted prime factors, challenging existing probabilistic heuristics and generalizing previous results.

Findings

Asymptotic formula contradicts probabilistic heuristics in certain ranges

Results extend to integers with prime factors in arithmetic progressions

Provides new insights into prime factor distributions in specified sets

Abstract

Let $E_0,\ldots,E_n$ be a partition of the set of prime numbers, and define $E_j(x) := \sum_{p \in E_j \atop p \leq x} \frac{1}{p}$. Define $\pi(x;\mathbf{E},\mathbf{k})$ to be the number of integers $n \leq x$ with $k_j$ prime factors in $E_j$ for each $j$. Basic probabilistic heuristics suggest that $x^{-1}\pi(x;\mathbf{E},\mathbf{k})$, modelled as the distribution function of a random variable, should satisfy a joint Poisson law with parameter vector $(E_0(x),\ldots,E_n(x))$, as $x \rightarrow \infty$. We prove an asymptotic formula for $\pi(x;\mathbf{E},\mathbf{k})$ which contradicts these heuristics in the case that for each $j$, $E_j(x)^2 \leq k_j \leq \log^{\frac{2}{3}-\epsilon} x$ for each $j$ under mild hypotheses. As a particular application, we prove an asymptotic formula regarding integers with prime factors from specific arithmetic progressions, which generalizes a result due to Delange.