The distribution of prime ideals of imaginary quadratic fields

Harman, G.; Kumchev, A.; Lewis, P.

doi:10.1090/S0002-9947-03-03104-0

The distribution of prime ideals of imaginary quadratic fields
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by G. Harman, A. Kumchev and P. A. Lewis PDF

Trans. Amer. Math. Soc. 356 (2004), 599-620 Request permission

Abstract:

Let $Q(x, y)$ be a primitive positive definite quadratic form with integer coefficients. Then, for all $(s, t)\in \mathbb R^2$ there exist $(m, n) \in \mathbb Z^2$ such that $Q(m, n)$ is prime and \[ Q(m - s, n - t) \ll Q(s, t)^{0.53} + 1. \] This is deduced from another result giving an estimate for the number of prime ideals in an ideal class of an imaginary quadratic number field that fall in a given sector and whose norm lies in a short interval.

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