A Montgomery-Hooley theorem for the k-fold divisor function

2023/02/21 by Tomos Parry, Parry, Tomos
Arts and Humanities · Mathematics · Social Sciences · #Analytic Number Theory Research #FOS: Mathematics #Historical Geopolitical and Social Dynamics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2302.11045

openalex publication_date 2023/02/21 · openalex created_date 2023/02/24 · openalex updated_date 2026/07/28

Abstract

Let d_k(n) denote the k-fold divisor function. For a wide range of large q the expected bound ∑_n≤ x\atop n≡ a(q)d_k(n)-\text main term ≈ √ x/q is shown to be true in an average sense -- for all k. This generalises the work of Pongsriiam and Vaughan [15] who studied k=2, and answers the work of Rodgers and Soundararajan [17], who used the asymptotic large sieve to study a smoothed version of the problem. We use a circle method approach as developed by Goldston and Vaughan [7] to study the unsmoothed problem.

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