2024/09/19 by Cheuk Fung Lau, Lau, Cheuk Fung
Mathematics · #11N05 #11N13 (Primary) 11N36 (Secondary) #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2409.12819
openalex publication_date 2024/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∈\mathbb N with q ≥ 3, at least mφ(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 60mlog 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 \lceillog m\rceil. A recursive combinatorial argument then shows that if q is squarefree and q ≫ (log m)2, then at least ≫ \fracm(log m)10 φ(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 ≫ (log m)2, then at least ≫ e-O(m log2 m/log m) φ(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 Erdős--Rankin type construction.