Pierre-Alexandre Bazin

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

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