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The weighted linear sieve and Selberg’s $$\lambda^ 2$$-method. (English) Zbl 0549.10036
Suppose that a set $$A=A_ X$$ of integers satisfies the properties usually assumed [cf. the book of H. Halberstam and H.-E. Richert, Sieve methods (1974; Zbl 0298.10026)] in the theory of the linear sieve. Thus the number of elements of A divisible by d is written in the form $$(X/d)\rho (d)+R(A,d),$$ where the function $$\rho$$ has average value 1 in the sense now usually assumed in this subject. The ”level of distribution” y is to be such that it is meaningful to work with $$d\leq y$$, as would be the case if (for example) we have $$\sum_{d\leq y}| \mu (d)R(A,d)| <X/\log^ 2X.$$ Let the ”degree” g be such that $$1<a<y^ g$$ for all a in A.
This paper continues a study of the question: for each integer $$R\geq 2$$, for what $$\Lambda_ R=R-\delta_ R$$ can we establish that if $$g<\Lambda_ R$$ then the set A contains a number having at most R prime factors, if X is sufficiently large. In a previous publication [Acta Arith. 40, 297-332 (1982; Zbl 0412.10033)] the author improved upon earlier results in this subject via a study of an expression of the form $$\sum_{d| a}\mu (d)\chi (d)\{1-\sum_{p| d}w(p)\}$$ where $$\chi$$ (d) was the function implicit in the sieve method of Rosser and Iwaniec [see e.g. H. Iwaniec’s paper, Acta Arith. 36, 171-202 (1980; Zbl 0435.10029)] to which this method would reduce if w(p) were replaced by zero.
In the present paper the function $$\chi$$ (d) is a slight modification of that implicit in the alternative treatment by W. B. Jurkat and H.-E. Richert [Acta Arith. 11, 217-240 (1965; Zbl 0128.269)] of the linear sieve problem (see also the extended version in the text by Halberstam and Richert cited above). It is shown that in the problem of the weighted linear sieve the two methods are not equivalent. Numerical computations indicate that the new method leads to improved bounds for $$\Lambda_ R$$ in the cases $$R=2,3,4$$. In particular we can now take $$\delta_ 2=0.044560.$$
The methods used involve a study of the effect of Selberg’s $$\lambda^ 2$$-method when considered as a weighted sieve, and lead to a qualitative theorem whose details are inappropriately long for inclusion in this review.

##### MSC:
 11N35 Sieves
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