The Pólya–Vinogradov inequality with log log q for every character

Let $\chi$ be a primitive Dirichlet character modulo $q\ge3$ and let $M(\chi)$ be the largest modulus of a partial sum $\sum_{n\le x}\chi(n)$. The Pólya–Vinogradov inequality bounds $M(\chi)$ by a constant times $\sqrt q\log q$. Paley showed that $\sqrt q\log\log q$ is the most that can be hoped for, and Montgomery and Vaughan proved that this is the truth if the generalised Riemann hypothesis holds. Without any hypothesis we prove that $M(\chi)\le C\sqrt q\log\log q$ for every primitive character $\chi$ to every modulus $q\ge3$, with an absolute constant $C$, and that $|L'/L(1,\chi)|\le C\log\log q$ for every nonprincipal character. We also prove, from the elementary bounds of Chebyshev alone, that for every $\varepsilon>0$ there is a constant $C_\varepsilon$ such that at most $q^{\varepsilon}$ of the primitive characters modulo $q$ have $M(\chi)>C_\varepsilon\sqrt q\log\log q$, and at most $q^{\varepsilon}$ of the nonprincipal characters have $|L'/L(1,\chi)|>C_\varepsilon\log\log q$. The proof has two parts. The first is a transfer theorem for a single character: if $\theta<1$ is fixed and the $L$-functions of the twists $\chi\psi$ of $\chi$ by the characters $\psi$ of the moduli up to $(\log q)^{16}$ have no zero in the rectangle $\Re s>\theta$, $|\Im s|\le5q^2$, then $M(\chi)\le C_\theta\sqrt q\log\log q$. It is proved by a contour shift for Riesz means of prime sums, a dissection of the circle into major and minor arcs, and a weak form of the Montgomery–Vaughan bound for exponential sums with multiplicative coefficients, which we prove with Selberg's sieve as the only arithmetic input. The second part counts the characters that fail the hypothesis of the transfer theorem by a zero-density theorem in the manner of Ingham and Montgomery, which bounds the number of zeros of all $L$-functions to a modulus in such a rectangle by a power of the modulus that tends to zero as $\theta$ tends to one. The same transfer theorem shows that a zero-free half-plane $\Re s>\theta$ for all Dirichlet $L$-functions gives the bound for every character, and such a half-plane exists: OpenAI has proved that every Dirichlet $L$-function, and every finite-order Hecke $L$-function over $\mathbb Q(\sqrt{-3})$, has no zero with $\Re s>11/12$, by a mean square of twisted Möbius sums over the Eisenstein integers, two Poisson summations, the transformation of Kubota's cubic theta function and the quadratic large sieve. Their proof is reproduced in full in a second appendix, with credit to them, so that the paper is complete in itself: the zero-density theorem is proved in the first appendix with explicit constants, and every statement of the paper is proved in the text.