The P\'olya--Vinogradov inequality with $\log\log q$
We prove that every primitive Dirichlet character $\chi$ modulo $q\ge3$ has character sums $\sum_{n\le x}\chi(n)$ of size at most $C\sqrt q\log\log q$, with an absolute constant $C$. Montgomery and Vaughan proved this bound under the generalised Riemann hypothesis, and Paley showed that it is optimal. The only input from outside the paper is the zero-free half-plane $\Re s>7/8$ for Dirichlet $L$-functions. On the major arcs this half-plane gives a power saving for twisted prime sums; on the minor arcs we prove a weak form of the Montgomery--Vaughan bound for exponential sums with…
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