Linear correlations amongst numbers represented by positive definite binary quadratic forms

TL;DR

This paper investigates linear correlations among representation functions of positive definite binary quadratic forms, deriving asymptotic formulas for sums involving products of these functions evaluated at linear shifts.

Contribution

It introduces new asymptotic results for linear correlations of representation functions of binary quadratic forms, expanding understanding of their distribution patterns.

Findings

Derived asymptotic formulas for sums of products of representation functions

Established linear correlation behavior among quadratic form representations

Enhanced theoretical understanding of quadratic form representation distributions

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 r_1, ..., r_k are representation functions, we obtain an asymptotic for sum_{n,d} r_1(n) r_2(n+d) ... r_k(n+ (k-1)d).