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

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 (2018), we do not require averaging over the moduli. Our proof adapts a technique of S. Lester (2016) who treated the variance of the $k$-fold divisor function in the short intervals setting in the same range, and is based on a smoothed Voronoï summation formula but twisted by multiplicative characters. The use of Dirichlet characters allows us to extend to a wider range than the previous result of Kowalski and Ricotta (2014) who used additive characters. Smoothing also permits us to treat all $k$ unconditionally. This result is closely related to moments of Dirichlet $L$-functions.