Almost primes in various settings

2018/06/23 by Lewulis, Paweł
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1806.09034

Abstract

Let k ≥ 3 and let L_i(n) = A_in + B_i be some linear forms such that A_i and B_i are integers. Define P(n) = ∏_i=1^k L_i(n). For each k it is known that Ω(P (n) ) ≤ ρ_k infinitely often for some integer ρ_k. We improve the possible values of ρ_k for 4 ≤ k ≤ 10 assuming GEH. We also show that we can take ρ₅=14 unconditionally. As a by-product of our approach we reprove the ρ₃=7 result which was previously obtained by Maynard who used techniques specifically designed for this case.

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