The variance of divisor sums in arithmetic progressions

摘要

We study the variance of sums of the k-fold divisor function d k ? ( n ) {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.