A disproof of Hooley's conjecture

Abstract

Define $G(x;q)$ to be the variance of primes $p\le x$ in the arithmetic progressions modulo $q$, weighted by $\log p$. Hooley conjectured that as soon as $q$ tends to infinity and $x\ge q$, we have the upper bound $G(x;q) \ll x \log q$. In this paper we show that the upper bound does not hold in general, and that $G(x;q)$ can be asymptotically as large as $x (\log q+\log\log\log x)^2/4$.