In number theory, Vinogradov's theorem is a result which implies that any sufficiently large odd integer can be written as a sum of three prime numbers. It is a weaker form of Goldbach's weak conjecture, which would imply the existence of such a representation for all odd integers greater than five. It is named after Ivan Matveyevich Vinogradov, who proved it in the 1930s. Hardy and Littlewood had shown earlier that this result followed from the generalized Riemann hypothesis, and Vinogradov was able to remove this assumption. The full statement of Vinogradov's theorem gives asymptotic bounds on the number of representations of an odd integer as a sum of three primes. The notion of "sufficiently large" was ill-defined in Vinogradov's original work, but in 2002 it was shown that 101346 is sufficiently large. Additionally numbers up to 1020 had been checked via brute force methods, thus only a finite number of cases to check remained before the odd Goldbach conjecture would be proven or disproven. In 2013, Harald Helfgott claimed to have proved Goldbach's weak conjecture for all cases.
Statement of Vinogradov's theorem Let A be a positive real number. Then
r ( N ) = 1 2 G ( N ) N 2 + O ( N 2 log − A N ) , {\displaystyle r(N)={1 \over 2}G(N)N^{2}+O\left(N^{2}\log ^{-A}N\right),}
where
r ( N ) = ∑ k 1 + k 2 + k 3 = N Λ ( k 1 ) Λ ( k 2 ) Λ ( k 3 ) , {\displaystyle r(N)=\sum _{k_{1}+k_{2}+k_{3}=N}\Lambda (k_{1})\Lambda (k_{2})\Lambda (k_{3}),}
using the von Mangoldt function Λ {\displaystyle \Lambda } , and
G ( N ) = ( ∏ p ∣ N ( 1 − 1 ( p − 1 ) 2 ) ) ( ∏ p ∤ N ( 1 + 1 ( p − 1 ) 3 ) ) . {\displaystyle G(N)=\left(\prod _{p\mid N}\left(1-{1 \over {\left(p-1\right)}^{2}}\right)\right)\left(\prod _{p\nmid N}\left(1+{1 \over {\left(p-1\right)}^{3}}\right)\right).}
A consequence If N is odd, then G(N) is roughly 1, hence N 2 ≪ r ( N ) {\displaystyle N^{2}\ll r(N)} for all sufficiently large N. By showing that the contribution made to r(N) by proper prime powers is O ( N 3 2 log 2 N ) {\displaystyle O\left(N^{3 \over 2}\log ^{2}N\right)} , one sees that
N 2 log − 3 N ≪ ( number of ways N can be written as a sum of three primes ) . {\displaystyle N^{2}\log ^{-3}N\ll \left({\hbox{number of ways N can be written as a sum of three primes}}\right).}
This means in particular that any sufficiently large odd integer can be written as a sum of three primes, thus showing Goldbach's weak conjecture for all but finitely many cases.
Strategy of proof The proof of the theorem follows the Hardy–Littlewood circle method. Define the exponential sum
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