Representations of integers by certain positive definite binary quadratic forms

Abstract

Abstract. We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n = x 2 + N y 2 for a squarefree integer N .

A remark on a conjecture of Borwein and Choi

“…The following result describes a relationship between genus characters χ and the orders of poles of L 2 (s, χ ⊗ χ). The proof is similar to that of Proposition 2.4 in [10].…”

“…Proof. As the proof is similar to that of Theorem 1.3 in [10], we sketch the relevant details. If −N ≡ 1 mod 4, then the discriminant of K = Q( √ −N ) is −N .…”

Estimates for representation numbers of quadratic forms

“…To our knowledge, the only (nontrivial) results on estimates/asymptotics of n≤x r f (n) β for generic discriminants D (in particular, with more than one form per genus) known so far are Bernays's result (see [2]), (1.14) for β = 0 in [3], and parts of (1.5) for fixed discriminant and β = 2 in [11]. We have seen that the question of obtaining asymptotics for N f (x) remains open in the intermediate range (see (1.2)), whereas we have now obtained asymptotics for positive integer moments of r f (n) in all ranges (see Corollary 1).…”