Residue Class Patterns of Consecutive Primes

Abstract

For $m,q \in \mathbb{N}$, we call an $m$-tuple $(a_1,\ldots,a_m) \in \prod_{i=1}^m (\mathbb{Z}/q\mathbb{Z})^\times$ good if there are infinitely many consecutive primes $p_1,\ldots,p_m$ satisfying $p_i \equiv a_i \pmod{q}$ for all $i$. We show that given any $m$ sufficiently large, $q$ squarefree, and $A \subseteq (\mathbb{Z}/q\mathbb{Z})^\times$ with $|A|=\lfloor 71(\log m)^3 \rfloor$, we can form at least one non-constant good $m$-tuple $(a_1,\ldots,a_m) \in \prod_{i=1}^m A$. Using this, we can provide a lower bound for the number of residue class patterns attainable by consecutive primes, and for $m$ large and $\varphi(q) \gg (\log m)^{10}$ this improves on the lower bound obtained from direct applications of Shiu (2000) and Dirichlet (1837). The main method is modifying the Maynard-Tao sieve found in Banks, Freiberg, and Maynard (2015), where instead of considering the 2nd moment we considered the $r$-th moment, where $r$ is an integer depending on $m$.