Residue Class Patterns of Consecutive Primes

Abstract

Dickson's conjecture and the Hardy--Littlewood prime tuple conjecture predict that every pattern of reduced residue classes modulo $q$ is attained by infinitely many strings of $m$ consecutive primes. At present, however, even proving that a single non-constant residue class pattern of length $m$ occurs infinitely often is beyond the reach of existing methods. Combining Dirichlet's theorem on primes in arithmetic progressions with a theorem of Shiu (2000) shows that, for any $m,q\in\mathbb N$ with $q \ge 3$, at least $m\varphi(q)$ residue class patterns of length $m$ are attained by infinitely many consecutive primes. In this paper, we prove that if $q$ is squarefree, every prescribed sequence of at least $60m\log m$ reduced residue classes mod $q$ contains, in order, an $m$-term block pattern that occurs infinitely often among consecutive primes, with each constant block of length at most $\lceil\log m\rceil$. A recursive combinatorial argument then shows that if $q$ is squarefree and $q \gg (\log m)^2$, then at least \[ \gg \frac{m}{(\log m)^{10}} \varphi(q)^2 \] residue class patterns of length $m$ occur infinitely often among consecutive primes. Moreover, we also show that if $q$ is squarefree and $q \gg (\log m)^2$, then at least \[ \gg e^{-O(m \log_2 m/\log m)} \varphi(q)^{m/\lceil \log m \rceil} \] residue class patterns of length $m$ occur infinitely often among consecutive primes. The proof consists of a modification of the Maynard--Tao sieve found in Banks, Freiberg, and Maynard (2016), by considering the $r$-th moment instead of the 2nd moment for an integer $r$ depending on $m$, which is then combined with an Erd\H{o}s--Rankin type construction.