On almost-prime k-tuples

2023/01/12 by Bin Chen, Chen, Bin
Mathematics · #11N05 #11N35 #11N36 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2301.05044

openalex publication_date 2023/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let τ denote the divisor function and H=\h₁,...,h_k\ be an admissible set. We prove that there are infinitely many n for which the product ∏_i=1^k(n+h_i) is square-free and ∑_i=1^kτ(n+h_i)≤ \lfloor ρ_k\rfloor, where ρ_k is asymptotic to (2126)/(2853) k². It improves a previous result of M. Ram Murty and A. Vatwani, replacing 2126/2853 by 3/4. The main ingredients in our proof are the higher rank Selberg sieve and Irving-Wu-Xi estimate for the divisor function in arithmetic progressions to smooth moduli.

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