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 [1]: 92–93 write: A curious feature of sieve literature is that while there is frequent use of Brun's method there are only a few attempts to formulate a general Brun theorem (such as Theorem 2.1); as a result there are surprisingly many papers which repeat in considerable detail the steps of Brun's argument.Diamond & Halberstam[2]: 42 attribute the terminology Fundamental Lemma to Jonas Kubilius.
We use these notations: This formulation is from Tenenbaum.
[4]: 60 Other formulations are in Halberstam & Richert,[1]: 82 in Greaves,[3]: 92 and in Friedlander & Iwaniec.
[5]: 732–733 We make the assumptions: There is a parameter
In the sieve it is related to the number of levels of the inclusion–exclusion principle.
The fundamental lemma has almost the same form as for the combinatorial sieve.
is no longer an independent parameter at our disposal, but is controlled by the choice of
Note that the error term here is weaker than for the fundamental lemma of the combinatorial sieve.
Halberstam & Richert remark:[1]: 221 "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."