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].

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].

Reviewer: S.W.Graham (Mt.Pleasant)