Distribution of primes of imaginary quadratic fields in sectors

The Hooley–Huxley Contour Method for Problems in Number Fields

“…To apply Theorem 1 note that When K=Q(i) Zarzycki has, in [16], given another application of the Hooley Huxley method. This time both the norm and argument of the ideals are equally constrained as in our Theorem 1.…”

“…This time both the norm and argument of the ideals are equally constrained as in our Theorem 1. Unfortunately, [16] lacks references to necessary results such as zero-free regions for Hecke L-functions which we hope the present paper furnishes. Also, the quality of the final results in [16] depends on zero density results such as (30) and there are too few details in the equivalent result, Lemma 2 of [16], to verify the quoted result.…”

“…Unfortunately, [16] lacks references to necessary results such as zero-free regions for Hecke L-functions which we hope the present paper furnishes. Also, the quality of the final results in [16] depends on zero density results such as (30) and there are too few details in the equivalent result, Lemma 2 of [16], to verify the quoted result. Further, the application in [16] to prime ideals in sectors can be dealt with by more classical methods, as in [3].…”

Effective Sato-Tate conjecture for abelian varieties and applications

“…The added ingredient in [JP17] is the use of unconditional estimates for the number of primes in an imaginary quadratic field lying in a sector; such an estimate has been given by Maknys [Mak83], modulo a correction described in [JP17]. (For the Gaussian integers, see also [Zar91]. )…”