Exponential sums over primes are unbounded

Abstract

We prove prime exponential sums have no better than square root cancellation on average on short intervals, in the sense that $$\frac{1}{x} \sum_{-y< n\le x} \left|\sum_{\substack{n< m \le n+y\\ 1\le m \le x}} \Lambda(m) \mathrm{e}(\alpha m)\right|^2 \gg y\log y$$ whenever $y \ll x^{1/3-\varepsilon}.$ This answers a question of Ramar\'e by proving the lower bound $$\sup_{n\le x} \left|\sum_{m\le n} \Lambda(m) \mathrm{e}(\alpha m)\right| \gg x^{1/6 - \varepsilon}.$$