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- W2809933404 abstract "Let k ⩾ 3 $k geqslant 3$ and let L i ( n ) = A i n + B i $L_i(n) = A_in + B_i$ be some linear forms such that A i $A_i$ and B i $B_i$ are integers. Define P ( n ) = ∏ i = 1 k L i ( n ) ${mathcal {P}(n) = prod _{i=1}^k L_i(n)}$ . For each k it is known that Ω ( P ( n ) ) ⩽ ρ k $Omega (mathcal {P} (n) ) leqslant rho _k$ infinitely often for some integer ρ k $rho _k$ . We improve the possible values of ρ k $rho _k$ for 4 ⩽ k ⩽ 10 $4 leqslant k leqslant 10$ assuming the generalized Elliot–Halberstam conjecture. We also show that we can take ρ 5 = 14 $rho _5=14$ unconditionally. As a by-product of our approach we reprove the ρ 3 = 7 $rho _3=7$ result which was previously obtained by Maynard who used techniques specifically designed for this case." @default.
- W2809933404 created "2018-07-10" @default.
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- W2809933404 date "2022-04-23" @default.
- W2809933404 modified "2023-09-26" @default.
- W2809933404 title "Almost primes in various settings" @default.
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- W2809933404 doi "https://doi.org/10.1112/mtk.12132" @default.
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