A lower bound for the variance of generalized divisor functions in\n arithmetic progressions

2020/04/12 by Daniele Mastrostefano, Mastrostefano, Daniele
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Primary: 11N64. Secondary: 11B25

paper · pdf · doi:10.48550/arxiv.2004.05602

openalex publication_date 2020/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for a large class of multiplicative functions, referred to as\ngeneralized divisor functions, it is possible to find a lower bound for the\ncorresponding variance in arithmetic progressions. As a main corollary, we\ndeduce such a result for any \α-fold divisor function, for any complex\nnumber \α not\∈ 1 \∪-\ℕ, even when considering a sequence\nof parameters \α close in a proper way to 1. Our work builds on that of\nHarper and Soundararajan, who handled the particular case of k-fold divisor\nfunctions d_k(n), with k\∈\ℕ_{\≥ 2}.\n

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