Around 1920, British mathematicians Hardy and Littlewood greatly developed analytic number theory and established powerful tools such as "circle method" to study number theory. They used the "circle method" in their paper published in 1923, and proved that under the premise of generalized Riemann conjecture, every odd number large enough can be expressed as the sum of three prime numbers, and almost every even number large enough can be expressed as the sum of two prime numbers [7][8].

1937 is a year of great breakthrough in the study of weak Goldbach conjecture. First of all, T. Eastman proved that every sufficiently large odd number can be expressed as the sum of two odd prime numbers and a number not exceeding the product of two prime numbers:

Or [5]

In the same year, the former Soviet mathematician Ivan Matveyevich vinogradov (ива? н Матве? евич Виногра? дов) On the basis of using the circle method, the dependence on Riemann conjecture in Hardy's and Littlewood's works is removed. That is to say, vinogradov proved that every odd number big enough can be expressed as the sum of three prime numbers, and almost every even number big enough can be expressed as the sum of two prime numbers. Vinogradov's proof uses his original method to estimate the exponential sum with prime numbers as variables in more detail, that is to say, in order to better divide the upper arc and the lower arc and directly estimate the integral on the lower arc, it can be ignored without using the generalized Riemann conjecture. The only drawback is that vinogradov did not give the lower limit of "large enough". Later, in 1956, Poirot sturgeon gave a computable lower limit:, which means that an integer greater than can be written as the sum of three prime numbers [13]. 1946, the former Soviet mathematician Linnik (ю? рий Влади? мирович Ли? нник) Along the path of Hardy and Littlewood, vinogradov's result was also proved by the method of function theory [13]. But the lower limit of vinogradov's theorem is still too large for practical application. There are 684,665,438+068 digits written. To verify that the previous even number can be written as the sum of two prime numbers, the amount of calculation is still too large. 1989, Chen Jingrun and Wang Yuan lowered the lower limit to [14], and in 5438+0 in 2006, Liao Mingzhe and Wang Tianze further lowered the lower limit to [9], but it was still far from the actual verification range (). If the generalized Riemann conjecture is assumed to be correct, J-M Deshuillers and others proved in 1998 that every odd number greater than or equal to 7 can be written as the sum of three prime numbers (that is, the complete proof of the weak Goldbach conjecture under the assumption that the generalized Riemann conjecture is correct) [15].

In 1938, Hua proved a generalization of weak Goldbach conjecture: given any integer k, every odd number large enough can be expressed as. When k = 1, it is a weak Goldbach conjecture [5].

Because the method used in vinogradov estimation is essentially the sieve method, mathematicians also hope to use an analytical method similar to the circle method instead. In 1945, Linnik developed a method to estimate the zero density of Dirichlet's L function, and used it to prove that the integral on the lower arc can be ignored, thus proving the weak Goldbach conjecture by pure analysis. This proof is very complicated. Later, several mathematicians put forward their own simplified proofs. In 1975, Vaughan proposed the first method to estimate the zero density of L function independently. In 1977, Pan Chengdong obtained a simple proof of the elementary properties only with L function [5].