On the Brun--Titchmarsh theorem

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 x satisfying p a q. We strength this inequality for q/ x in different ranges, improving upon previous works by Motohashi, Goldfeld, Iwaniec, Friedlander and Iwaniec, and Maynard for general or special moduli. In particular, we obtain a Burgess-like constant in the full range q<x^1/2-. The proof is based on various estimates for character and exponential sums, by appealing to arithmetic exponent pairs and bilinear forms with algebraic trace functions from -adic cohomology, and sums of Kloosterman sums from spectral theory of automorphic forms, as well as large value theorem for Dirichlet polynomials due to Huxley.