Linear correlations amongst numbers represented by positive definite binary quadratic forms
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, ..., rk are representation functions, we obtain an asymptotic for sumn,d r1(n) r2(n+d) ... rk(n+ (k-1)d).
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