On the sieve parameters \(\alpha_\kappa\) and \(\beta_\kappa\) for large \(\kappa\)
The general sieve of \textit{N. C. Ankeny} and \textit{H. Onishi} [Acta Arith. 10, 31-62 (1964; Zbl 0127.27002)] uses the Selberg sieve and one or two iterations of Buchstab's identity. In a long series of papers, Diamond, Halberstam, and Richert developed a sieve method that combines Selberg sieves and an optimal number of iterations of Buchstab's identity. This paper discusses two parameters that occur in the sieves of Diamond, Halberstam, and Richert (DHR sieves). For each sieve dimension \(\kappa > 1\), the parameter \(\alpha_\kappa \) denotes the point beyond which the DHR upper bound sieve improves on the Ankeny-Onishi sieve. The other parameter is the sieving limit \(\beta_\kappa\), the point beyond which the DHR lower sieve yields a nontrivial lower bound. The authors show that for all large enough \(\kappa\), \(\alpha_\kappa\) lies within a bounded distance of \(\rho_\kappa\), the largest positive zero of Iwaniec's function \(q_\kappa\). They also show that for all \(\kappa \geq 200\), \(\beta_\kappa\) is smaller than \(\nu_\kappa\), the sieving limit of the Ankeny-Onishi sieve.
scientific article; zbMATH DE number 1097548
scientific article