Fundamental Lemma of Sieve Theory - Fundamental Lemma of The Selberg Sieve

Fundamental Lemma of The Selberg Sieve

This formulation is from Halberstam & Richert. Another formulation is in Diamond & Halberstam.

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:

• w(p) / p < 1 - c for some small fixed c and all p
• | R_d | ≤ ω(d) where ω(d) is the number of distinct prime divisors of d.

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."