On an Erd\H{o}s--Kac-type conjecture of Elliott

TL;DR

This paper investigates an Erdős–Kac-type conjecture by Elliott related to the distribution of the number of prime factors in shifted primes, connecting it to the Bombieri–Vinogradov theorem and Poisson–Dirichlet distribution.

Contribution

The paper proves that Elliott's conjecture follows from the Bombieri–Vinogradov theorem and establishes a related result involving the Poisson–Dirichlet distribution using advanced prime distribution results.

Findings

Elliott's conjecture is a consequence of the Bombieri–Vinogradov theorem.

A new result involving the Poisson–Dirichlet distribution is proved.

Deeper level of prime distribution results are employed in the analysis.

Abstract

Elliott and Halberstam proved that $\sum_{p<n} 2^{\omega(n-p)}$ is asymptotic to $\phi(n)$. In analogy to the Erd\H{o}s--Kac Theorem, Elliott conjectured that if one restricts the summation to primes $p$ such that $\omega(n-p)\le 2 \log \log n+\lambda(2\log \log n)^{1/2}$ then the sum will be asymptotic to $\phi(n)\int_{-\infty}^{\lambda} e^{-t^2/2}dt/\sqrt{2\pi}$. We show that this conjecture follows from the Bombieri--Vinogradov Theorem. We further prove a related result involving Poisson--Dirichlet distribution, employing deeper lying level of distribution results of the primes.