On an Erdős--Kac-type conjecture of Elliott

Abstract

Elliott and Halberstam proved that ${\sum }_{p<n}{2}^{\omega (n-p)}$ is asymptotic to $\varphi (n)$. In analogy to the Erdős--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 $\varphi (n){\int }_{-\infty }^{\lambda }{e}^{-t^2/2}dt/\sqrt{2\pi }$. 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.

16 pages, accepted version. Title shortened, typos fixed, two remarks added

References in corpus (1)

• Multiplicative arithmetic functions and the generalized Ewens measure