A Bombieri-Vinogradov Theorem for primes in short intervals and small sectors

TL;DR

This paper extends the Bombieri-Vinogradov theorem to primes represented by prime ideals in number fields within short intervals and small sectors, demonstrating their even distribution in arithmetic progressions.

Contribution

It introduces a new distribution result for primes in number fields within short intervals and sectors, generalizing prior work by Duke and Coleman.

Findings

Primes in number fields are well-distributed in short intervals and sectors.

Extension of Bombieri-Vinogradov theorem to primes in algebraic number fields.

Primes satisfy small sector conditions with respect to Hecke characters.

Abstract

Let $K$ be a finite Galois extension of $\mathbb{Q}$. We count primes in short intervals represented by the norm of a prime ideal of $K$ satisfying a small sector condition determined by Hecke characters. We also show that such primes are well-distributed in arithmetic progressions in the sense of Bombieri-Vinogradov. This extends previous work of Duke and Coleman.