Strings of congruent primes in short intervals II

TL;DR

This paper establishes a lower bound on the frequency of prime pairs with small gaps and specific congruence conditions, improving previous bounds by Shiu, contributing to understanding prime distributions in short intervals.

Contribution

It provides a new lower bound for primes with small gaps and fixed residue classes, advancing the knowledge of prime patterns in short intervals.

Findings

Improves upon Shiu's bound for primes in specific residue classes

Establishes a lower bound for primes with small gaps and congruence conditions

Enhances understanding of prime distribution in short intervals

Abstract

Let $p_1 = 2, p_2 = 3,...$ be the sequence of all primes. Let $\epsilon$ be an arbitrarily small but fixed positive number, and fix a coprime pair of integers $q \ge 3$ and $a$. We will establish a lower bound for the number of primes $p_r$, up to $X$, such that both $p_{r+1} - p_{r} < \epsilon \log p_r$ and $p_{r} \equiv p_{r+1} \equiv a \bmod q$ simultaneously hold. As a lower bound for the number of primes satisfying the latter condition, the bound we obtain improves upon a bound obtained by D. Shiu.