as where is a density, meaning a multiplicative function such that and is an approximation of and is some remainder term. The sifting function becomes or in short One tries then to estimate the sifting function by finding upper and lower bounds for respectively and . The partial sum of the sifting function alternately over- and undercounts, so the remainder term will be huge. Brun's idea to improve this was to replace in the sifting function with a weight sequence consisting of restricted Möbius functions. Choosing two appropriate sequences and and denoting the sifting functions with and , one can get lower and upper bounds for the original sifting functions Since is multiplicative, one can also work with the identity Notation: a word of caution regarding the notation, in the literature one often identifies the set of sequences with the set itself. This means one writes to define a sequence . Also in the literature the sum is sometimes notated as the cardinality of some set , while we have defined to be already the cardinality of this set. We used to denote the set of primes and for the greatest common divisor of and . Types of sieving Modern sieves include the Brun sieve, the Selberg sieve, the Turán sieve, the large sieve, and the larger sieve. One of the original purposes of sieve theory was to try to prove conjectures in number theory such as the twin prime conjecture. While the original broad aims of sieve theory still are largely unachieved, there have been some partial successes, especially in combination with other number theoretic tools. Highlights include: Brun's theorem, which shows that the sum of the reciprocals of the twin primes converges (whereas the sum of the reciprocals of all primes diverges); Chen's theorem, which shows that there are infinitely many primes p such that p + 2 is either a prime or a semiprime (the product of two primes); a closely related theorem of Chen Jingrun asserts that every sufficiently large even number is the sum of a prime and another number which is either a prime or a semiprime. These can be considered to be near-misses to the twin prime conjecture and the Goldbach conjecture respectively. The fundamental lemma of sieve theory, which asserts that if one is sifting a set of N numbers, then one can accurately estimate the number of elements left in the