ALMOST-PRIME $k$-TUPLES. Issue 1 (6th September 2013) - Maynard, James
Abstract: Let $k\geq 2$ and $\Pi (n)= { \mathop{\prod }\nolimits}_{i= 1}^{k} ({a}_{i} n+ {b}_{i} )$ for some integers ${a}_{i}, {b}_{i} $ ( $1\leq i\leq k$ ). Suppose that $\Pi (n)$ has no fixed prime divisors. Weighted sieves have shown for infinitely many integers $n$ that the number of prime factors $\Omega (\Pi (n))$ of $\Pi (n)$ is at most ${r}_{k} $, for some integer ${r}_{k} $ depending only on $k$ . We use a new kind of weighted sieve to improve the possible values of ${r}_{k} $ when $k\geq 4$ .