Exponential sums over primes are unbounded

We prove prime exponential sums have no better than square root cancellation on average on short intervals, in the sense that $$\frac{1}{x}\sum _{-y<n\le x}{\mid \sum _{\begin{array}{c}n<m\le n+y\\ 1\le m\le x\end{array}}\mathrm{\Lambda }(m)\mathrm{e}(\alpha m)\mid }^2\gg y\log y$$ whenever $y\ll x^{1\mathrm{/}3-\epsilon }\mathrm{.}$ This answers a question of Ramaré by proving the lower bound $$\underset{n\le x}{\sup }\mid \sum _{m\le n}\mathrm{\Lambda }(m)\mathrm{e}(\alpha m)\mid \gg x^{1\mathrm{/}6-\epsilon }\mathrm{.}$$