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: Fundamental lemma of the combinatorial sieve. 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."