On the Restricted Divisor Function in Arithmetic Progressions

2010/03/28 by Igor E. Shparlinski, Shparlinski, Igor E.
Mathematics · #11N37 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.NT #msc:11N37

paper · pdf · doi:10.48550/arxiv.1003.5347

openalex publication_date 2010/03/28 · arxiv created 2010/08/04 · arxiv updated 2010/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain several asymptotic estimates for the sums of the restricted divisor function τ_M,N(k) = #\1 ≤ m ≤ M, 1≤ n ≤ N: mn = k\ over short arithmetic progressions, which improve some results of J. Truelsen. Such estimates are motivated by the links with the pair correlation problem for fractional parts of the quadratic function αk², k=1,2,... with a real α.

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