Residue classes containing an unexpected number of primes

TL;DR

This paper investigates specific residue classes modulo q that contain fewer primes than expected, revealing unexpected prime distribution patterns in arithmetic progressions for certain fixed residues.

Contribution

It demonstrates that for particular fixed residues, the distribution of primes in arithmetic progressions deviates significantly from the expected average.

Findings

Certain residue classes contain fewer primes than predicted

The deviation is significant for specific fixed residues

Results contribute to understanding prime distribution irregularities

Abstract

We fix a non-zero integer $a$ and consider arithmetic progressions $a \bmod q$, with $q$ varying over a given range. We show that for certain specific values of $a$, the arithmetic progressions $a \bmod q$ contain, on average, significantly fewer primes than expected.