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 $$\frac{1}{x}\sum _{-y<n\le x}{\left|\sum _{\begin{array}{c}n<m\le n+y\\ 1\le m\le x\end{array}}\Lambda (m)\mathrm{e}(\alpha m)\right|}^2\gg y\log y$$ whenever $y\ll x^{1/3-\epsilon }.$ This answers a question of Aymone and Ramaré by proving the lower bound $$\underset{n\le x}{\sup }\left|\sum _{m\le n}\Lambda (m)\mathrm{e}(\alpha m)\right|\gg x^{1/6-\epsilon }.$$

11 pages