A Bombieri-Vinogradov theorem for exponential sums over products of k primes

Abstract

We prove a Bombieri-Vinogradov type theorem for exponential sums over products of $k$ primes. As an application, we show the lower bound $$\underset{n\le x}{\sup }\left|\sum _{m\le n}{1}_{\Omega (m)=k}e(\alpha m)\right|\gg x^{1/6-\epsilon }$$ for $2\le k\le (2-\epsilon )\log \log x$ and $\alpha \in \mathbb{R},$ where we noted $e(\beta ):=e^{2i\pi \beta }.$

14 pages