Linear correlations amongst numbers represented by positive definite binary quadratic forms

Details Linear correlations amongst numbers represented by positive definite binary quadratic forms Linear correlations amongst numbers represented by positive definite binary quadratic forms

2011-06-23

arxiv.org/abs/1106.4690v2

Mathematics

Number Theory

60 pages. Revised and extended version

Scientific paper

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).

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