On the Brun--Titchmarsh theorem. II

Denote by $\pi (x;q,a)$ the number of primes $p⩽x$ with $p\equiv aq\mathrm{.}$ We prove new upper bounds for $\pi (x;q,a)$ when $q$ is a large prime very close to $\sqrt{x}$, improving upon the classical work of Iwaniec (1982). The proof reduces to bounding a quintilinear sum of Kloosterman sums, to which we introduce a new shifting argument inspired by Vinogradov–Burgess–Karatsuba, going beyond the classical Fourier-analytic approach thanks to a deep algebro-geometric result of Kowalski–Michel–Sawin on sums of products of Kloosterman sums.