Almost-prime $k$-tuples

Abstract

Let $k\ge 2$ and $\Pi (n)={\prod }_{i=1}^k(a_{i}n+b_i)$ for some integers $a_i,b_i$ ($1\le i\le k$). Suppose that $\Pi (n)$ has no fixed prime divisors. Weighted sieves have shown for infinitely many integers $n$ that $\Omega (\Pi (n))\le r_k$ 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 $k\ge 4$.

26 pages