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Some applications of sieves of dimension exceeding 1. (English) Zbl 0941.11034
Greaves, G. R. H. (ed.) et al., Sieve methods, exponential sums, and their applications in number theory. Proceedings of a symposium, Cardiff, UK, July 17-21, 1995. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 237, 101-107 (1997).
In Chapter 10 of the book [Sieve Methods, London Mathematical Society Monographs 4. Academic Press (1974; Zbl 0444.10038)], H. Halberstam and H.-E. Richert described a general weighted sieve procedure for use in problems involving sieve dimension larger than 1. More recently, the authors, together with H.-E. Richert, [J. Number Theory 28, 306-346 (1988; Zbl 0639.10031)] have presented new upper and lower bound sieves for dimension $$\kappa > 1$$. In this article, the authors combine these sieves with the above-mentioned weighted sieve procedure. This yields improved estimates in applications such as almost-primes in polynomial sequences.
To illustrate their results, we mention one example. Let $$H_i(n)$$ be a linear polynomial for $$i=1,\dots, r$$, so that $$H_i(n)=a_i n + b_i$$ for some integers $$a_i$$ and $$b_i$$. Let $$H(n)= \prod_{i=1}^n H_i(n)$$, and suppose that $$H(n)$$ has no fixed prime divisors. Suppose also that $$\prod_{i=1}^g a_i \prod_{1\leq r < t \leq g} (a_r b_t- a_t b_r) \neq 0$$. It is possible to prove that there exists an $$r=r(g)$$ such that there are infinitely many integers $$n$$ for which $$H(n)$$ has at most $$r$$ prime factors. In [Sieve Methods (loc. cit.], Halberstam and Richert established this with $$r(2)=6, r(3)=10, r(4)=15,$$ and $$r(5)= 19$$. In this article, the same result is proved with $$r(2)=5, r(3)=8, r(4)=12,$$ and $$r(5)=16.$$ The authors remark that they just missed getting $$r(2)=4$$; “a more delicate weighting procedure, or a better 2-dimensional sieve, should give $$r=4.$$”
A novel application given at the end of the paper answers a question raised at the 1993 West CoastNumber Theory Conference. “For which $$r$$ are there only finitely many primitive Pythagorean triples $$(x,y,z)$$ such that the number of distinct prime factors of $$xyz$$ is $$r$$?” Kevin Ford has observed that there are only 30 such triples with $$r\leq 5$$. The authors apply the results developed here to show that there are infinitely many triples with $$r\leq 17$$. Ford argues heuristically there are probably infinitely many such triples with $$r=6$$; this argument is given at the end of the paper.
For the entire collection see [Zbl 0910.00038].

##### MSC:
 11N36 Applications of sieve methods 11N13 Primes in congruence classes