arXiv subjects
Publications and source records attributed to Péa Bazin.
We prove a Bombieri-Vinogradov type theorem for exponential sums over products of $k$ primes. As an application, we show the lower bound $$\sup_{n\le x} \left|\sum_{m\le n} 1_{Ω(m) = k} e(αm)\right| \gg x^{1/6 - \varepsilon}$$ for $2\le k\le (2-\varepsilon)\log\log x$ and $α\in\mathbb{R},$ where we noted $e(β) := e^{2iπβ}.$
We prove prime exponential sums have no better than square root cancellation on average on short intervals, in the sense that $$\frac{1}{x} \sum_{-y< n\le x} \left|\sum_{\substack{n< m \le n+y\\ 1\le m \le x}} Λ(m) \mathrm{e}(αm)\right|^2 \gg y\log y$$ whenever $y \ll x^{1/3-\varepsilon}.$ This answers a question of Aymone and Ramaré by proving the lower bound $$\sup_{n\le x} \left|\sum_{m\le n} Λ(m) \mathrm{e}(αm)\right| \gg x^{1/6 - \varepsilon}.$$