On the Sieve Parametersακandβκfor Largeκ

✍ Scribed by Harold G Diamond; H Halberstam

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 484 KB
• Volume: 67
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1997.2166

✦ Synopsis

dedicated to the memory of our friend, paul erdo s Two parameters, : } and ; } , play a central role in the sieve method of Diamond, Halberstam, and Richert. For each value of the sieve dimension }>1, : } is the point beyond which the DHR upper sieve improves upon the upper bound sieve function of Ankeny and Onishi; and the sieving limit ; } is the point beyond which the DHR lower sieve yields a nontrivial (i.e., positive) lower bound. We show that for all large enough }, : } lies within a bounded distance of \ } , the largest positive zero of Iwaniec's function q } ( } ) and that ; } is smaller than & } , the sieving limit of the Ankeny Onishi sieve.

1997 Academic Press

The authors, in association with the late Professor H.-E. Richert, have constructed a class of sieves for all dimensions }>1 (see [DHR1 DHR9]) that are hybrids of Selberg's upper bound sieve and a variation of the combinatorial sieves of Rosser Iwaniec ([I]). Some applications of these sieves to problems of dimension not exceeding 10 are described in [DH1], and from these it is apparent that the DHR method often improves somewhat on the sieves of Ankeny and Onishi (AO sieves for short) [AO] (see also Chap. 10]).

The DHR method involves two parameters of crucial importance: for }>1, : } is the point beyond which the DHR upper sieve improves upon that of Ankeny and Onishi; and the sieving limit ; } beyond which the DHR lower sieve yields a nontrivial (i.e., positive) lower bound.

In this article we do two things: we show that for all large enough }, : } lies within a bounded distance of \ } , the largest positive zero of Iwaniec's function q } ( } ) (see the q-section of the Appendix for references); and that article no. NT972166 52 0022-314XÂ97 25.00