DISTRIBUTION OF PRIMES OF IMAGINARY QUADRATIC FIELDS IN SECTORS

Let π(x; φ1, φ2; β, γ) be the number of primes p from ℤ such that p≡β (mod γ), N(p)≤x, φ1≤arg p≤φ2. We prove that for sufficiently large x. π (x + y ; φ{symbol}1, φ{symbol}2 ; β, y) - π (x ; φ{symbol}1, φ{symbol}2 ; β, y) ∼ frac(4 y (φ{symbol}2 - φ{symbol}1), 2 π φ{symbol} (y) log x). if only x2/3+ε≤y=o(x) and φ2-φ1≫x(-1/3)+ε. This improves Maknys' result which is 11/16+ε≤y=o(x) and φ2-φ1≫x(-5/16)+ε. © 1991 Academic Press, Inc.