Principal forms X^2 + nY^2 representing many integers

Abstract

In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X^2+nY^2. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n=2. In this paper, we prove that in fact this constant is unbounded as n runs through positive integers with a fixed number of prime divisors.

10 pages, title has been changed, Sections 2 and 3 are new, to appear in Abh. Math. Sem. Univ. Hamburg