The variance of divisor sums in arithmetic progressions

TL;DR

This paper investigates the variance of divisor sums in arithmetic progressions, confirming a conjecture in a specific range and linking results to moments of Dirichlet L-functions using the large sieve.

Contribution

It provides the first averaged asymptotic results for the variance of divisor sums in sparse arithmetic progressions, connecting to L-function moments.

Findings

Confirmed an averaged conjecture about variance asymptotics

Established connections between divisor sums and L-function moments

Utilized the asymptotic large sieve method

Abstract

We study the variance of sums of the $k$-fold divisor function $d_k(n)$ over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture about the asymptotics of this variance. This result is closely related to moments of Dirichlet $L$-functions and our proof relies on the asymptotic large sieve.