The distribution of the variance of primes in arithmetic progressions

Abstract

Hooley conjectured that the variance V(x;q) of the distribution of primes up to x in the arithmetic progressions modulo q is asymptotically x log q, in some unspecified range of q\leq x. On average over 1\leq q \leq Q, this conjecture is known unconditionally in the range x/(log x)^A \leq Q \leq x; this last range can be improved to x^{\frac 12+\epsilon} \leq Q \leq x under the Generalized Riemann Hypothesis (GRH). We argue that Hooley's conjecture should hold down to (loglog x)^{1+o(1)} \leq q \leq x for all values of q, and that this range is best possible. We show under GRH and a linear independence hypothesis on the zeros of Dirichlet L-functions that for moderate values of q, \phi(q)e^{-y}V(e^y;q) has the same distribution as that of a certain random variable of mean asymptotically \phi(q) log q and of variance asymptotically 2\phi(q)(log q)^2. Our estimates on the large deviations of this random variable allow us to predict the range of validity of Hooley's Conjecture.