Sums of three primes

Abstract

Vinogradov proved that every sufficiently large odd integer is the sum of three primes. In addition, he obtained an asymptotic fromula for the number of representations of an odd integer as the sum of three prime numbers. Vinogradov’s theorem is one of the great results in additive prime number theory. The principal ingredients of the proof are the circle method and an estimate of a certain exponential sum over prime numbers.

The method which I discovered in 1937 for estimating sums over primes permits, in the first instance, the evaluation of an estimate for the simplest of such sums, i.e. a sum of the type: EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeaaca % WGLbWaaWbaaSqabeaacaaIYaGaeqiWdaNaamyAaiabeg7aHjaadcha % aaaabaGaamiCaiabgsMiJkaad6eaaeqaniabggHiLdaaaa!429B!EquationSource$$ \sum\limits_{p \leqslant N} {{e^{2\pi i\alpha p}}} $$.

This estimation in combination with the previoulsy known theorems concerning the distribution of primes in arithmetic progressions... paved the way for establishing unconditionally the asymptotic formula of Hardy and Littlewood in the Goldbach ternary representation problem. I. M. Vinogradov [135, page 365]