ALMOST-PRIME $k$-TUPLES. Issue 1 (6th September 2013) - Maynard, James
Abstract: Let $k≥ 2$ and $Π (n)= { \mathop{\prod }\nolimits}_{i= 1}^{k} ({a}_{i} n+ {b}_{i} )$ for some integers ${a}_{i}, {b}_{i} $ ( $1≤ i≤ k$ ). Suppose that $Π (n)$ has no fixed prime divisors. Weighted sieves have shown for infinitely many integers $n$ that the number of prime factors $Ω (Π (n))$ of $Π (n)$ 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 $k≥ 4$ .