How to get a lower bound of the number of numbers left after a sieve?

Question

Is there any (simple) way to get a more usefull bound than >=0 on the number of numbers left after a/any sieve like the sieve of eratosthenes? What material should one read to get knowledge of this?

Answer 1

It depends on what you mean by simple. Each sieve is different, and different methods apply. The short answer is no --- for instance, one cannot deduce a nontrivial lower bound on an application of the Sieve of Eratosthenes when applied to twim primes. [Doing so would lead to a proof of the twin prime conjecture].

A good book for figuring out how sieves work is Sieve Methods by Halberstam. The first sieve considered in detail is the Sieve of Eratosthenes and its various applications.