Sifting Limits for the \Lambda^2\Lambda^- Sieve

TL;DR

This paper computes sifting limits for the Lambda^2 Lambda^- sieve across various dimensions, demonstrating its superiority over other sieves for dimensions greater than or equal to 3, and introduces a method for calculating sieve functions.

Contribution

It provides new sifting limit calculations for the Lambda^2 Lambda^- sieve and compares its effectiveness to other sieves, along with outlining a method for computing sieve functions.

Findings

Lambda^2 Lambda^- sieve outperforms other sieves for κ ≥ 3

Sifting limits are computed for 1<κ≤10

A method for computing sieve functions for integral κ is outlined

Abstract

Sifting limits for the $\Lambda^{2}\Lambda^{-}$ sieve, Selberg's lower bound sieve, are computed for integral dimensions $1<\kappa\le10$. The evidence strongly suggests that for all $\kappa\ge3$ the $\Lambda^{2}\Lambda^{-}$ sieve is superior to the competing combinatorial sieves of Diamond, Halberstam, and Richert. A method initiated by Grupp and Richert for computing sieve functions for integral $\kappa$ is also outlined.