Gaussian Primes in Narrow Sectors

TL;DR

This paper extends previous results by counting Gaussian primes within narrow sectors, considering both their norm and argument restrictions, thus advancing understanding of their distribution in the complex plane.

Contribution

It generalizes Ricci's theorem to include Gaussian primes with simultaneous restrictions on norm and argument within short intervals.

Findings

Established asymptotic counts for Gaussian primes in narrow sectors.

Demonstrated the distribution of Gaussian primes under combined norm and argument constraints.

Extended the theoretical framework for prime distribution in algebraic number fields.

Abstract

We generalize a Theorem of Ricci and count Gaussian primes $\mathfrak{p}$ with short interval restrictions on both the norm and the argument of $\mathfrak{p}$.