On the regularity of primes in arithmetic progressions

TL;DR

This paper investigates the distribution irregularities of primes in arithmetic progressions within specific intervals, generalizing a conjecture and extending results to prime ideals in number fields.

Contribution

It generalizes Recaman's conjecture and demonstrates non-uniform distribution of primes in certain intervals, including prime ideals in number fields.

Findings

Primes do not distribute uniformly in certain intervals among residue classes.

Generalization of Recaman's conjecture to broader contexts.

Extension of results to prime ideals in number fields.

Abstract

We prove that for a positive integer $k$ the primes in certain kinds of intervals can not distribute too 'uniformly' among the reduced residue classes modulo $k$. Hereby, we prove a generalization of a conjecture of Recaman and establish our results in a much more general situation, in particular for prime ideals in number fields.