Vinogradov's theorem

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.

Let A be a positive real number. Then

where

using the von Mangoldt function ${\displaystyle \Lambda }$, and

If N is odd, then G(N) is roughly 1, hence ${\displaystyle N^{2}\ll r(N)}$ for all sufficiently large N. By showing that the contribution made to r(N) by proper prime powers is ${\displaystyle O\left(N^{3 \over 2}\log ^{2}N\right)}$, one sees that

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