2025/09/05 by Le Duc Hieu, Hieu, Le Duc
Mathematics · #11B30 #11N13 Secondary: 11N36 #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT) #Primary: 11N05
paper · pdf · doi:10.48550/arxiv.2509.04883
openalex publication_date 2025/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that once θ>17/30, every sufficiently long interval [x,x+xθ] contains many k-term arithmetic progressions of primes, uniformly in the starting point x. More precisely, for each fixed k≥3 and θ>17/30, for all sufficiently large X and all x∈[X,2X], #\k-APs of primes in [x,x+xθ]\ ≫k,θ \fracN2((φ(W)/W)k(log R)k) \asymp \fracX2θ(log X)k+1+o(1), where W:=∏p≤ \tfrac12loglog Xp, N:=\lfloor xθ/W\rfloor, and R:=Nη for a small fixed η=η(k,θ)>0. This is obtained by combining the uniform short-interval prime number theorem at exponents θ>17/30 (a consequence of recent zero-density estimates of Guth and Maynard) with the Green-Tao transference principle (in the relative Szemerédi form) on a window-aligned W-tricked block. We also record a concise Maynard-type lemma on dense clusters restricted to a fixed congruence class in tiny intervals (log x)ε, which we use as a warm-up and for context. An appendix contains a short-interval Barban-Davenport-Halberstam mean square bound (uniform in x) that we use as a black box for variance estimates. The proofs in this paper were assisted by GPT-5.