Linear correlations amongst numbers represented by positive definite binary quadratic forms

2011/06/23 by Lilian Matthiesen, Matthiesen, Lilian · 2 citations
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1106.4690

60 pages. Small corrections

openalex publication_date 2011/06/23 · arxiv created 2012/06/19 · arxiv updated 2012/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a positive definite binary quadratic form f, let r(n) = |(x,y): f(x,y)=n| denote its representation function. In this paper we study linear correlations of these functions. For example, if r₁, ..., r_k are representation functions, we obtain an asymptotic for sum_n,d r₁(n) r₂(n+d) ... r_k(n+ (k-1)d).

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