On the Brun--Titchmarsh theorem. I

The classical Brun--Titchmarsh theorem gives an upper bound, which is of correct order of magnitude in the full range, for the number of primes $p\leqslant x$ satisfying $p\equiv a\bmod q$. We strengthen this inequality for different ranges of $\log q/\log x$, improving upon previous works by Motohashi, Goldfeld, Iwaniec, Friedlander and Iwaniec, and Maynard for general or special moduli. In particular, we are able to beat Iwaniec's barrier $q<x^{9/20-}$, and improve all existing inequalities in the range $x^{9/20}\ll q<x^{1/2-}$ by utilizing bilinear or trilinear structures in the remainder terms of linear sieve. The proof is based on various estimates for character and exponential sums, which we derive by appealing to arithmetic exponent pairs and bilinear forms with algebraic trace functions from $\ell$-adic cohomology, trilinear forms with Kloosterman fractions, and sums of Kloosterman sums from spectral theory of automorphic forms, as well as large value theorem for Dirichlet polynomials.