Distribution of primes of imaginary quadratic fields in sectors (Q753861)

The author investigates the asymptotic distribution of Gaussian primes \(\pi\in \mathbb Z[i]\) satisfying \(x-y<N\pi \leq x\), \(\pi\equiv \beta mod \gamma\) and \(\phi_ 1\leq \arg \pi \leq \phi_ 2\). The stated formula improves a result obtained previously by \textit{M. Maknys} [Acta Math. Hung. 42, 131--138 (1983; Zbl 0525.10026)]. The theme of that paper was the problem about the distance between consecutive prime ideals of imaginary quadratic fields in sectorial domains. The proof of the main theorem is based on a new form of contour integration, introduced by Huxley and Hooley. Unfortunately, some of the fundamental tools are only sketched. Using ideas of a paper of \textit{M. D. Coleman} [Proc. Lond. Math. Soc., III. Ser. 61, 433--456 (1990; Zbl 0712.11065)], it seems to be possible to improve the corollaries.