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.
Statement of Vinogradov's theorem
Let A be a positive real number. Then
using the von Mangoldt function , and
If N is odd, then G(N) is roughly 1, hence for all sufficiently large N. By showing that the contribution made to r(N) by proper prime powers is , one sees that
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. In 2013, Harald Helfgott proved Goldbach's weak conjecture for all cases.
Strategy of proof
Then we have
where denotes the number of representations restricted to prime powers . Hence
If is a rational number , then can be given by the distribution of prime numbers in residue classes modulo . Hence, using the Siegel-Walfisz theorem we can compute the contribution of the above integral in small neighbourhoods of rational points with small denominator. The set of real numbers close to such rational points is usually referred to as the major arcs, the complement forms the minor arcs. It turns out that these intervals dominate the integral, hence to prove the theorem one has to give an upper bound for for contained in the minor arcs. This estimate is the most difficult part of the proof.
If we assume the Generalized Riemann Hypothesis, the argument used for the major arcs can be extended to the minor arcs. This was done by Hardy and Littlewood in 1923. In 1937 Vinogradov gave an unconditional upper bound for . His argument began with a simple sieve identity, the resulting terms were then rearranged in a complicated way to obtain some cancellation. In 1977 R. C. Vaughan found a much simpler argument, based on what later became known as Vaughan's identity. He proved that if , then
Using the Siegel-Walfisz theorem we can deal with up to arbitrary powers of , using Dirichlet's approximation theorem we obtain on the minor arcs. Hence the integral over the minor arcs can be bounded above by
which gives the error term in the theorem.
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