When the number of divisors is a quadratic residue

2017/01/09 by Olivier Bordellès, Bordellès, Olivier
Computer Science · Mathematics · #11M41 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Primary 11N37 #Secondary 11A25 #math.NT #msc:11A25 #msc:11M41 #msc:11N37

paper · pdf · doi:10.48550/arxiv.1701.02286

9 pages

arxiv created 2017/01/09 · openalex publication_date 2017/01/09 · arxiv updated 2017/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let q > 2 be a prime number and define λ_q := ( \fracτq ) where τ(n) is the number of divisors of n and ( (⋅)/(q) ) is the Legendre symbol. When τ(n) is a quadratic residue modulo q, then ( λ_q ⋆ 1 ) (n) could be close to the number of divisors of n. This is the aim of this work to compare the mean value of the function λ_q ⋆ 1 to the well known average order of τ. The proof reveals that the results depend heavily on the value of ( (2)/(q) ). A bound for short sums in the case q=5 is also given, using profound results from the theory of integer points close to certain smooth curves.

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