Sieve Methods And Their Applications

Sieve theory, is a famous method to sifter primes in number theory. More precisely, it’s uesd for siftering the elements of a sequence (?) which are coprime to a function P(z). Here P(z) is a truncated function associated with (?) (cf. [16]). Sieve methods maybe derive from Eratosthene in the period of ancient Greece, but the sieve of Eratosthene has some limitation in practice because of the error term, the choice of parameter and so on; Later, the Norwegian mathematician Brun created Brun’s combinatorial sieve between 1917 and 1924, and Brun’s sieve is well practical, we can get good results in many problems; In 1947, based on an optimal design of quadratic form, Selberg created the Selberg sieve. Compared with Brun’s sieve, it’s more concise in the theory, more flexible in the skills and more precise in the estimation; So far the best reslut of combinatorial sieve is the Rosser-Iwaniec sieve in 1980’s [18] [19], which is considered as the extreme of sieve methods. Not only it’s more flexible in dealing with the error term, but also it may have an improvement of the constant in estimating the upper bound of the main term(cf. [26]).Sieve methods are powerful tools in number theory, which are applicative in many well-known problems, such as the Goldbach conjecture, the twin prime conjecture, the conjecture that there exist infinitely many integers such that an irreducible polynomial is prime, and so on. In 1973, based on the work of Bombieri [2], Renyi [24], Pan Chengdong [21] and Wang Yuan [27], Chen Jingrun [7] proved the famous "1+2" Chen’s theorem by using the double sieve, i.e. if x is a sufficiently large even integer, p is a prime, and denote by P2 as any integer having at most two prime factors, equal or distinct, then the equation x=p+P2 is solvable. And let Px(1,2) be the number of solutions to the above equation, then we haveIf we can improve P2 to a prime p, then we get the Goldbach conjecture. As for the twin conjecture, we denote pn as the n-th prime, andIn 1940, Erdos [11] first proved A< 1 by Brun’s sieve. In 2009, Goldston, Pintz and Yildirim [14] first proved A= 0 by using sieve, and later in 2010, they [15] provedNow we also call the sieve method that they used as the sieve of G-P-Y. Then in 2014, Zhang Yitang [30] achieved a breakthrough in the twin prime conjecture. Based on the work of Goldston, Pintz and Yildirim, he provedIn the process of improving the above two famous conjectures, sieve methods all play a very important role. Moreover, sieve methods is also very useful for the problem of the largest prime factors of consecutive integers. If n, n+1 are integers, we denote F(n), P(n+1) as the largest prime factors of n, n+1 respectively andWe conjecture that when x is sufficiently large, E(x) is asymptotically equal to x/2, and more generally, the largest prime factors of n and n+1 are"inde-pendent events". This conjecture seems very simple, but it’s a very difficulty problem. The famous mathematician Tenenbaum [25] has said "It lies in the same class of problems than the famous abc-conjecture". The first reslut in this aspect maybe come from Erdos and Pomerance [13], who in 1978 proved the following result:as E(x) is defined above, for a sufficient large x, we have E(x)≥0.0099x. But the sieve methods do not play a crucial role in their proof. In 2005, de la Breteche, Pomerance and Tenenbaum[4] get the estimation by sieve methodsEspecially, in the end of the article the authors said that later Fouvry indicated that we can get a better result by using a more sophisticated sieve method, but they do not give the process of the proof.In the second chapter of this article,we will introduce some basic con-ception of sieve methods:the sequences (?), the sifting set (?) and the sifting function S((?)). We will also explain the essential idea of sieve methods when introducing the sifting function. In the third chapter, we will introduce some kinds of sieve methods:the sieve of Eratosthene-Legendre, Brun’s com-binatorial sieve, the Selberg sieve and the Rosser-Iwaniec sieve. And in the fourth chapter, we will give two applications of sieve methods. First, in or-der to master sieve methods and the choices of the sequences and the sieve functions better, we will give a brief proof of Chen’s theorem. Second, for the problem of the largest prime factors of consecutive integers, we will give the full proof that Fouvry indicated above, which can improve the lower bound to E(x)≥0.05866x.