Almost-prime $k$-tuples

Let $k\ge 2$ and $Π(n)=\prod_{i=1}^k(a_in+b_i)$ for some integers $a_i, b_i$ ($1\le i\le k$). Suppose that $Π(n)$ has no fixed prime divisors. Weighted sieves have shown for infinitely many integers $n$ that $Ω(Π(n))\le r_k$ holds for some integer $r_k$ which is asymptotic to $k\log{k}$. We use a new kind of weighted sieve to improve the possible values of $r_k$ when $k\ge 4$.

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