Vinogradov's theorem

Let A be a positive real number. Then

${\displaystyle r(N)={1 \over 2}G(N)N^{2}+O\left(N^{2}\log ^{-A}N\right),}$

where

${\displaystyle r(N)=\sum _{k_{1}+k_{2}+k_{3}=N}\Lambda (k_{1})\Lambda (k_{2})\Lambda (k_{3}),}$

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

${\displaystyle G(N)=\left(\prod _{p\mid N}\left(1-{1 \over {\left(p-1\right)}^{2}}\right)\right)\left(\prod _{p\nmid N}\left(1+{1 \over {\left(p-1\right)}^{3}}\right)\right).}$

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

${\displaystyle N^{2}\log ^{-3}N\ll \left({\hbox{number of ways N can be written as a sum of three primes}}\right).}$

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.

The proof of the theorem follows the Hardy–Littlewood circle method. Define the exponential sum

${\displaystyle S(\alpha )=\sum _{n=1}^{N}\Lambda (n)e(\alpha n)}$.

Then we have

${\displaystyle S(\alpha )^{3}=\sum _{n_{1},n_{2},n_{3}\leq N}\Lambda (n_{1})\Lambda (n_{2})\Lambda (n_{3})e(\alpha (n_{1}+n_{2}+n_{3}))=\sum _{n\leq 3N}{\tilde {r}}(n)e(\alpha n)}$,

where ${\displaystyle {\tilde {r}}}$ denotes the number of representations restricted to prime powers ${\displaystyle \leq N}$. Hence

${\displaystyle r(N)=\int _{0}^{1}S(\alpha )^{3}e(-\alpha N)\;d\alpha }$.

If ${\displaystyle \alpha }$ is a rational number ${\displaystyle {\frac {p}{q}}}$, then ${\displaystyle S(\alpha )}$ can be given by the distribution of prime numbers in residue classes modulo ${\displaystyle q}$. 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 ${\displaystyle S(\alpha )}$ for ${\displaystyle \alpha }$ 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 ${\displaystyle |S(\alpha )|}$. 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 ${\displaystyle |\alpha -{\frac {a}{q}}|<{\frac {1}{q^{2}}}}$, then

${\displaystyle |S(\alpha )|\ll \left({\frac {N}{\sqrt {q}}}+N^{4/5}+{\sqrt {Nq}}\right)\log ^{4}N}$.

Using the Siegel–Walfisz theorem we can deal with ${\displaystyle q}$ up to arbitrary powers of ${\displaystyle \log N}$, using Dirichlet's approximation theorem we obtain ${\displaystyle |S(\alpha )|\ll {\frac {N}{\log ^{A}N}}}$ on the minor arcs. Hence the integral over the minor arcs can be bounded above by

${\displaystyle {\frac {CN}{\log ^{A}N}}\int _{0}^{1}|S(\alpha )|^{2}\;d\alpha \ll {\frac {N^{2}}{\log ^{A-1}N}}}$,

which gives the error term in the theorem.

• Vinogradov, Ivan Matveevich (1954). The Method of Trigonometrical Sums in the Theory of Numbers. Translated, revised and annotated by K. F. Roth and Anne Davenport. London and New York: Interscience. MR 0062183.
• Nathanson, Melvyn B. (1996). Additive number theory. The classical bases. Graduate Texts in Mathematics. Vol. 164. New York: Springer-Verlag. doi:10.1007/978-1-4757-3845-2. ISBN 0-387-94656-X. MR 1395371. Chapter 8.

• Weisstein, Eric W. "Vinogradov's Theorem". MathWorld.