Exponential sums over primes are unbounded

TL;DR

This paper establishes that exponential sums over primes do not exhibit cancellation better than the square root on short intervals, providing new lower bounds and answering a question posed by Ramaré.

Contribution

It proves a lower bound on prime exponential sums, showing they are unbounded and do not have better than square root cancellation on short intervals.

Findings

Prime exponential sums are unbounded on short intervals.

Lower bounds for prime exponential sums are established.

Answers a question of Ramaré regarding the size of these sums.

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 $$\sup_{n\le x} \left|\sum_{m\le n} \Lambda(m) \mathrm{e}(\alpha m)\right| \gg x^{1/6 - \varepsilon}.$$