Variants Of the Selberg Sieve and Almost Prime K-tuples

Abstract

Abstract Let $k\geqslant 2$ and $\mathcal{P} (n) = (A_1 n + B_1 ) \cdots (A_k n + B_k)$ where all the $A_i, B_i$ are integers. Suppose that $\mathcal{P} (n)$ has no fixed prime divisors. For each choice of k it is known that there exists an integer ϱk such that $\mathcal{P} (n)$ has at most ϱk prime factors infinitely often. We used a new weighted sieve setup combined with a device called ɛ-trick to improve the possible values of ϱk for $k\geqslant 7$. As a by-product of our approach, we improve the conditional possible values of ϱk for $k\geqslant 4$, assuming the generalized Elliott–Halberstam conjecture.