The fundamental lemma of sieve theory says that upper and lower sieve theory estimates approach the expected local-density product when the available distribution level is a sufficiently large power of the sifting limit. One formulation uses a finite set of positive integers andHere ranges over squarefree integers up to , is multiplicative with , and the local densities satisfy a fixed-dimension boundThen a standard version yieldsThe fundamental lemma of sieve theory is useful only after both errors are controlled: large reduces the relative main-term error, while the summed remainders measure the actual arithmetic distribution. It then guarantees survivors, often almost primes, without claiming they are prime.[ref]
Some sieve theory by Terence Tao (254A Notes 4). Equation (22), Lemma 17 and Corollary 19; endpoint convention uses primes below z
Articles by others on the same topic
The Fundamental Lemma of Sieve Theory is a key result in analytic number theory that underpins the operation of sieve methods, which are techniques used to count or estimate the size of sets of integers that have certain properties, particularly those related to prime numbers. The lemma essentially provides a way to bound the sums that arise in these sieve methods.