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arXiv · 2205.02354

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

David T. Nguyen
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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.

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David T. Nguyen. 2022-05-04. Variance of the k-fold divisor function in arithmetic progressions for individual modulus. https://arxiv.org/abs/2205.02354

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