The Pólya–Vinogradov inequality with $\log\log q$

We prove unconditionally that every primitive Dirichlet character $\chi$ modulo $q\ge3$ has character sums of size at most $C\sqrt q\log\log q$. Montgomery and Vaughan proved this bound under the generalised Riemann hypothesis, and Paley showed that it is optimal. The proof uses the zero-free half-plane $\Re s>7/8$, which also gives $|L'/L(1,\chi)|\ll\log\log q$ and two-sided bounds for the Euler–Kronecker constants of the cyclotomic fields $\mathbb Q(\zeta_q)$.