[Paper Review] The distribution of the variance of primes in arithmetic progressions

Daniel Fiorilli|arXiv (Cornell University)|Jan 23, 2013

Analytic Number Theory Research30 references3 citations

TL;DR

This paper proposes that Hooley's conjecture $ V(x;q) \sim x\log q $ holds for all $ q \geq (\log\log x)^{1+\delta} $, extending the known range under GRH and a linear independence hypothesis. Using probabilistic modeling of $ \phi(q)e^{-y}V(e^y;q) $, it shows this variance behaves like a random variable with mean $ \phi(q)\log q $ and variance $ 2\phi(q)(\log q)^2 $, predicting the asymptotic behavior down to $ q = (\log\log x)^{1+\delta} $, which is optimal.

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.

Motivation & Objective

• To determine the optimal range of $ q $ for which Hooley's conjecture $ V(x;q) \sim x\log q $ holds.
• To extend the known asymptotic results on the average of $ V(x;q) $ down to smaller values of $ q $, beyond the current conditional and unconditional bounds.
• To investigate the distribution of $ V(x;q) $ for individual $ q $, especially in the range $ q \leq x^{1/2} $, where no asymptotic results are known.
• To provide a probabilistic justification for the conjectured lower bound $ q \geq (\log\log x)^{1+\delta} $, showing this is best possible.
• To explore the transition in behavior of $ V(x;q) $ as $ q $ decreases, particularly near $ q \sim \log\log x $, using large deviation estimates.

Proposed method

• Model $ \phi(q)e^{-y}V(e^y;q) $ as a random variable under GRH and a linear independence hypothesis on the zeros of Dirichlet $ L $-functions.
• Use the limiting distribution of the vector $ (e^{i\gamma_1 y}, \dots, e^{i\gamma_k y}) $ to analyze equidistribution and derive the distribution of $ V(x;q) $.
• Apply large deviation estimates for the random variable $ H_q $, defined as the normalized version of $ \phi(q)e^{-y}V(e^y;q) $, to predict the range where $ V(x;q) \sim x\log q $.
• Derive the asymptotic mean $ \phi(q)\log q $ and variance $ 2\phi(q)(\log q)^2 $ for $ H_q $, showing convergence to a normal-like distribution.
• Use the tail behavior of $ H_q $ to estimate the smallest $ y $ such that $ |e^{-y}V(e^y;q) - \log q| \neq o(\log q) $, leading to the prediction $ y \approx \exp(c\phi(q)) $.
• Compare the predicted behavior with known bounds: for fixed $ q $, $ V(x;q) \ll x(\log\log\log x)^4 $, indicating a transition near $ q = (\log\log x)^{1+\delta} $.

Experimental results

Research questions

• RQ1 Can Hooley's conjecture $ V(x;q) \sim x\log q $ be extended to $ q \geq (\log\log x)^{1+\delta} $, and is this range optimal?
• RQ2 What is the probabilistic distribution of $ \phi(q)e^{-y}V(e^y;q) $ under GRH and linear independence of $ L $-function zeros?
• RQ3 How do large deviations of the random variable $ H_q $ inform the range of validity of $ V(x;q) \sim x\log q $?
• RQ4 What transition occurs in the size of $ V(x;q) $ as $ q $ decreases from $ x^{o(1)} $ to fixed values?
• RQ5 Can the known upper bound $ V(x;q) \ll x(\log\log\log x)^4 $ for fixed $ q $ be reconciled with the asymptotic $ V(x;q) \sim x\log q $ in the range $ q \geq (\log\log x)^{1+\delta} $?

Key findings

• Under GRH and a linear independence hypothesis, $ \phi(q)e^{-y}V(e^y;q) $ has a limiting distribution with mean asymptotically $ \phi(q)\log q $ and variance asymptotically $ 2\phi(q)(\log q)^2 $.
• The largest deviations of $ H_q $ occur when $ \epsilon \asymp 1 $, suggesting a phase transition in the behavior of $ V(x;q) $ near $ q \sim \log\log x $.
• The range $ q \geq (\log\log x)^{1+\delta} $ is predicted to be optimal for $ V(x;q) \sim x\log q $, as $ e^{-y}V(e^y;q) \sim \log q $ only holds in this range.
• For $ q \leq (\log\log x)^{1-\delta} $, the asymptotic $ V(x;q) \sim x\log q $ is expected to fail, indicating a sharp threshold.
• For fixed $ q $, the limiting distribution of $ e^{-y}V(e^y;q) $ has double-exponentially decaying tails, leading to the bound $ V(x;q) \ll x(\log\log\log x)^4 $.
• The estimate $ V(x;q) = x\mathcal{L}(q)\left(1 + O\left(\Psi(x)\sqrt{\frac{\log\log x}{\phi(q)}}\right)\right) $ holds for $ q \geq (\log\log x)^{1+\delta} $, with $ \mathcal{L}(q) = \log q - \gamma - \log(2\pi) - \sum_{p\mid q}\frac{\log p}{p-1} $, and $ \Psi(x) \to \infty $.