Disproving 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 $\log p$. In analogy with his $q$-analogue of Selberg's upper bound on the variance of primes in intervals, Hooley conjectured that as soon as $q$ tends to infinity and $x\ge q$, we have the upper bound $G(x;q) \ll x \log q$. This conjecture was proven true over function fields by Keating and Rudnick, using equidistribution results of Katz. In this paper we show that the upper bound does not hold in general, and that $G(x;q)$ can be much larger than $x\log q$ for values of $q$ which are $\asymp \log\log x$. This implies that a conjecture of the first author on the range of validity of Hooley's conjecture is essentially best possible.