On almost prime k-tuples

Abstract

Let k ⩾ 2 and a i , b i ( 1 ⩽ i ⩽ k ) be integers such that a i > 0 and ∏ 1 ⩽ i < j ⩽ k ( a i b j − a j b i ) ≠ 0 . Let Ω ( m ) denote the total number of prime factors of m. Suppose ∏ ( n ) : = ∏ i = 1 k ( a i n + b i ) has no fixed prime divisors. Results of the form # { n ⩽ x : Ω ( ∏ ( n ) ) ⩽ r k } ≫ x ( log x ) − k where r k is asymptotic to k log k have been obtained by using sieve methods, in particular weighted sieves. In this paper, we use another kind of weighted sieve due to Selberg to obtain improved admissible values for r k .