On the Sieve Parametersακandβκfor Largeκ

AbstractTwo parameters,ακandβκ, play a central role in the sieve method of Diamond, Halberstam, and Richert. For each value of the sieve dimensionκ>1,ακis the point beyond which the DHR upper sieve improves upon the upper bound sieve function of Ankeny and Onishi; and the sieving limitβκis the point beyond which the DHR lower sieve yields a nontrivial (i.e., positive) lower bound. We show that for all large enoughκ,ακlies within a bounded distance ofρκ, the largest positive zero of Iwaniec's functionqκ(·) and thatβκis smaller thanνκ, the sieving limit of the Ankeny–Onishi sieve