Variants of the Selberg sieve, and almost prime k-tuples

Abstract

Let $k\ge 2$ 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 ${\varrho }_k$ such that $\mathcal{P}(n)$ has at most ${\varrho }_k$ prime factors infinitely often. We used a new weighted sieve set-up combined with a device called an $\epsilon$-trick to improve the possible values of ${\varrho }_k$ for $k\ge 7$. As a by-product of our approach, we improve the conditional possible values of ${\varrho }_k$ for $k\ge 4$, assuming the generalized Elliott--Halberstam conjecture.

34 pages