Divisor problems and the pair correlation for the fractional parts of n²α

2009/08/30 by Jimi Lee Truelsen, Truelsen, Jimi Lee
Mathematics · #11K31 #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Number Theory (math.NT) #Primary 11B05 #Secondary 11J54 #math.NT #msc:11B05 #msc:11J54 #msc:11K31

paper · pdf · doi:10.48550/arxiv.0908.4389

27 pages

arxiv created 2009/08/30 · openalex publication_date 2009/08/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Z. Rudnick and P. Sarnak have proved that the pair correlation for the fractional parts of n² α is Poissonian for almost all α. However, they were not able to find a specific α for which it holds. We show that the problem is related to the problem of determining the number of (a,b,r) ∈ \N³ such that a ≤ M, b ≤ N, r ≤ K and p ab ≡ r (q) for p and q coprime. With suitable assumptions on the relative size of K, M, N and q one should expect there to be KMN/q such triples asymptotically and we will show that this holds on average.

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