Lower bounds for the variance of sequences in arithmetic progressions: primes and divisor functions

TL;DR

This paper introduces a new method to establish lower bounds on the variance of sequences like primes and divisor functions in arithmetic progressions, improving understanding without relying on unproven hypotheses.

Contribution

It develops a general approach to lower bound variance in arithmetic progressions, applying it to primes and divisor functions, surpassing previous conditional results.

Findings

Lower bound of (1-ε) QN log(Q^2/N) for the von Mangoldt function variance.

Lower bound of Q N (log N)^{k^2 - 1} for divisor functions d_k(n).

Results hold over broad ranges of Q relative to N.

Abstract

We develop a general method for lower bounding the variance of sequences in arithmetic progressions mod $q$, summed over all $q \leq Q$, building on previous work of Liu, Perelli, Hooley, and others. The proofs lower bound the variance by the minor arc contribution in the circle method, which we lower bound by comparing with suitable auxiliary exponential sums that are easier to understand. As an application, we prove a lower bound of $(1-\epsilon) QN\log(Q^2/N)$ for the variance of the von Mangoldt function $(\Lambda(n))_{n=1}^{N}$, on the range $\sqrt{N} (\log N)^C \leq Q \leq N$. Previously such a result was only available assuming the Riemann Hypothesis. We also prove a lower bound $\gg_{k,\delta} Q N (\log N)^{k^2 - 1}$ for the variance of the divisor functions $d_k(n)$, valid on the range $N^{1/2+\delta} \leq Q \leq N$, for any natural number $k \geq 2$.