Strings of congruent primes in short intervals II

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 $p_r\equiv p_{r+1}\equiv a\mathrm{mod}q$ 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.