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ALMOST-PRIME $k$-TUPLES

Part of: Multiplicative number theory

Published online by Cambridge University Press:  06 September 2013

James Maynard
James Maynard*
Affiliation:
Mathematical Institute, 24–29 St Giles’, Oxford, OX1 3LB, U.K. email maynard@maths.ox.ac.uk

Abstract

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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 $k\geq 4$.

MSC classification

Secondary: 11N05: Distribution of primes 11N35: Sieves 11N36: Applications of sieve methods

Information

Type
Research Article
Information
Mathematika , Volume 60 , Issue 1 , January 2014 , pp. 108 - 138
Copyright
Copyright © University College London 2013 

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