ALMOST-PRIME $k$-TUPLES. Issue 1 (6th September 2013)

• Record Type: Journal Article
• Authors: Maynard, James
• Is Part Of: Mathematika. Volume 60:Issue 1(2014)
• Page Start: 108
• Page End: 138
• Publication Date: 2013-09-06
• Subjects: 11N05, -- 11N35, -- 11N36 (primary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/S0025579313000028
• Languages: English
• ISSNs: 0025-5793

Abstract

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 $k\geq 4$ .