The variance of divisor sums in arithmetic progressions

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.

30 pages. Incorporates referee suggestions

Cited by in corpus (4)

• A combinatorial identity and the finite dual of infinite dihedral group algebra
• Generalized divisor functions in arithmetic progressions: I
• The variance of the number of sums of two squares in ${\mathbb{F}}_q[T]$ in short intervals
• Averages of long Dirichlet polynomials