The Pólya–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 multiplicative coefficients, with Selberg's sieve as the only arithmetic input. The same half-plane gives $|L'/L(1,\chi)|\ll\log\log q$ for every nonprincipal $\chi$ and, for every prime $q$, the two-sided bound $-C\log q\log\log q\le\gamma_q\le C\log q$ for the Euler–Kronecker constant of the cyclotomic field $\mathbb Q(\zeta_q)$.