Consecutive primes in short intervals

TL;DR

This paper establishes a lower bound on the number of consecutive primes within short intervals that are all congruent to a fixed residue modulo q, highlighting patterns in prime distributions.

Contribution

It provides a new lower bound for the count of consecutive primes in short intervals sharing the same residue class, advancing understanding of prime clustering.

Findings

Lower bound for primes in short intervals with fixed residue

Quantitative results on prime clustering behavior

Enhanced understanding of prime distribution patterns

Abstract

We obtain a lower bound for \[ \#\{x/2< p_{n}\leq x:\ p_n \equiv\ldots\equiv p_{n+m}\equiv a\text{ (mod $q$)},\ p_{n+m} - p_{n}\leq y\}, \] where $p_{n}$ is the $n^{\text{th}}$ prime.