The distribution of the variance of primes in arithmetic progressions

TL;DR

This paper investigates the distribution of primes in arithmetic progressions, proposing a conjecture about the variance that extends previous results and predicts its validity across a broader range of moduli.

Contribution

It extends Hooley's conjecture to smaller moduli and provides probabilistic models and large deviation estimates under GRH and zero independence hypotheses.

Findings

Hooley's conjecture likely holds for q down to (loglog x)^{1+o(1)}.

Under GRH, the distribution of the variance matches a specific random variable.

Large deviation estimates support the predicted range of validity.

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.