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:
- w(d) is a multiplicative function.
- The sifting density κ satisfies, for some constant C and any real numbers η and ξ with 2 ≤ η ≤ ξ:
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.
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