Fundamental lemma of sieve theory


In number theory, the fundamental lemma of sieve theory is any of several results that systematize the process of applying sieve methods to particular problems. Halberstam & Richert
write:
Diamond & Halberstam
attribute the terminology Fundamental Lemma to Jonas Kubilius.

Common notation

We use these notations:
This formulation is from Tenenbaum. Other formulations are in Halberstam & Richert, in Greaves,
and in Friedlander & Iwaniec.
We make the assumptions:
There is a parameter u ≥ 1 that is at our disposal. We have uniformly in A, X, z, and u that
In applications we pick u to get the best error term. In the sieve it represents the number of levels of the inclusion–exclusion principle.

Fundamental lemma of the Selberg sieve

This formulation is from Halberstam & Richert. Another formulation is in Diamond & Halberstam.
We make the assumptions:
The fundamental lemma has almost the same form as for the combinatorial sieve. Write u = ln X / ln z. The conclusion is:
Note that u is no longer an independent parameter at our disposal, but is controlled by the choice of z.
Note that the error term here is weaker than for the fundamental lemma of the combinatorial sieve. Halberstam & Richert remark: "Thus it is not true to say, as has been asserted from time to time in the literature, that Selberg's sieve is always better than Brun's."