The Brun-Hooley sieve
The sieve of the title is a version of Brun's sieve that is simpler than the full-blown version but is nevertheless nearly as powerful. In the paper [Acta Arith. 66, 359-368 (1994; Zbl 0821.11046)] of \textit{C. Hooley} it was called an almost pure sieve, and may be viewed as an elaboration of the simplest ``pure sieve'' of V. Brun. In Hooley's paper its application to the twin almost prime problem was discussed, but a more significant application was to a new result involving sums of two squares [J. Reine Angew. Math. 452, 79-109 (1994; Zbl 0799.11034)], where the combination of simplicity and power of the new sieve was essential to its successful application. The new method is most easily applied in the upper bound sifting situation, and (as in some other sieve methods) a more involved step is needed to derive a corresponding lower bound method. In the paper under review the authors replace Hooley's version of this step by an alternative and possibly simpler argument. They use it to derive a version of the twin almost primes theorem that is qualitatively as good as those given by Brun's method. A dual form of their method is also presented. They suggest that this may be used to improve a certain principle producing a new lower bound sieve from a combination of precursor sieves. The principle in question was used in the vector sieve of \textit{J. Brüdern} and \textit{E. Fouvry} [Compos. Math. 102, No. 3, 337-355 (1996; Zbl 0860.11057)], and appears to have been first enunciated by \textit{A. Selberg} [Proc. Symp. Pure Math. 20, 311-351 (1971; Zbl 0222.10048)]. The authors ask whether an improved version of the vector sieve might follow.
scientific article; zbMATH DE number 1443345