Vinogradov-Goldbach theorem

A theorem on the representation of all sufficiently large odd numbers by a sum of three prime numbers. It is a consequence of the asymptotic formula for the number $I(N)$ of solutions of the equation

$$p_1+p_2+p_3=N$$

in prime numbers, demonstrated by I.M. Vinogradov in 1937:

$$I(N)=\frac{N^2}{2r^3}S(N)+O\left(\frac{N^2}{r^{3.5-\epsilon}}\right),$$

where $N$ is odd, $r=\log N$ and

$$S(N)=\prod_p\left(1+\frac{1}{(p-1)^3}\right)\prod_{p|N}\left(1-\frac{1}{p^2-3p+3}\right)>0.6.$$

Cf. Vinogradov method; Goldbach problem.

References

[1]	I.M. Vinogradov, "Selected works" , Springer (1985) (Translated from Russian)
[2]	L.-K. Hua, "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen , 1 : 2 (1959) (Heft 13, Teil 1)