On an Erdős--Kac-type conjecture of Elliott

Elliott and Halberstam proved that ${\sum }_{p<n}{2}^{\omega (n-p)}$ is asymptotic to $\phi (n)$. In analogy to the Erdős–Kac Theorem, Elliott conjectured that if one restricts the summation to primes $p$ such that $\omega (n-p)\le 2\log \log n+\lambda (2\log \log n{)}^{1\mathrm{/}2}$ then the sum will be asymptotic to $\phi (n){\int }_{-\mathrm{\infty }}^{\lambda }{e}^{-t^2\mathrm{/}2}dt\mathrm{/}\sqrt{2\pi }$. 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.