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, 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.