Variance of the k-fold divisor function in arithmetic progressions for individual modulus

TL;DR

This paper proves a smoothed version of a conjecture on the variance of the k-fold divisor function in arithmetic progressions for individual composite moduli, extending previous results without requiring averaging over moduli.

Contribution

It introduces a new technique using smoothed Voronoi summation twisted by multiplicative characters to analyze variance unconditionally for all k.

Findings

Confirmed a smoothed variance conjecture for individual moduli

Extended the range of previous results using multiplicative characters

Related the variance analysis to moments of Dirichlet L-functions

Abstract

In this paper, we confirm a smoothed version of a recent conjecture on the variance of the k-fold divisor function in arithmetic progressions to individual composite moduli, in a restricted range. In contrast to a previous result of Rodgers and Soundararajan, we do not require averaging over the moduli. Our proof adapts a technique of S. Lester who treated in the same range the variance of the k-fold divisor function in the short intervals setting, and is based on a smoothed Voronoi summation formula but twisted by multiplicative characters. The use of Dirichlet characters allows us to extend to a wider range from previous result of Kowalski and Ricotta who used additive characters. Smoothing also permits us to treat all k unconditionally. This result is closely related to moments of Dirichlet L-functions.