ALMOST‐PRIME k‐TUPLES. Issue 1 (6th September 2013) - Maynard, James
Abstract: Let k ≥ 2 and Π ( n ) = ∏ 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 .