On an ErdŐs–Kac-Type Conjecture of Elliott

Elliott and Halberstam proved that $\sum_{p \lt n} 2^{\omega(n-p)}$ 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 $\omega(n-p)\le 2 \log \log n+\lambda(2\log \log n)^{1/2}$ then the sum will be asymptotic to $\phi(n)\int_{-\infty}^{\lambda} \mathrm{e}^{-t^2/2}\,\mathrm{d}t/\sqrt{2\pi}$. 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.