On the Brun--Titchmarsh theorem. II

Ping Xi, Junren Zheng

Denote by $\pi (x;q,a)$ the number of primes $p⩽x$ with $p\equiv a\mathrm{mod}q.$ We prove new upper bounds for $\pi (x;q,a)$ when $q$ is a large prime very close to $\sqrt{x}$, impro…