Primes and prime ideals in short intervals

TL;DR

This paper extends Cramér's short intervals theorem to primes in arithmetic progressions and prime ideals, assuming the Riemann Hypothesis, using the inertia property of prime counting functions.

Contribution

It provides a uniform proof of prime distribution in short intervals for both primes and prime ideals under the Riemann Hypothesis, generalizing existing results.

Findings

Proves an analog of Cramér's theorem for primes in arithmetic progressions

Extends results to prime ideals in number fields

Results are uniform across different structures

Abstract

We prove the analog of Cram\'er's short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann Hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based mainly on the inertia property of the counting functions of primes and prime ideals.