A New Proof on the Three Primes Theorem

(A) By the circle method of Hardy and Littlewood and his method on the estimation of trigonometrical sum with prime variables, I. M. Vinogradov [l] first proved in 1937 that every large odd number is the sum of three prime numbers which is usually called the Goldbach-Vinogradov theorem or the three primes theorem. Later, Ju. V. Linnik [2] and N. G. Tchudakov [3] gave another two proofs on this theorem based on the estimation of the density of zeros of L-functions. Recently, H. L. Montgomery [4] and M. N. Huxley [5] gave two simplified proofs which are also based on the estimation of the density of zeros of L-functions, and in their proofs, the approximate functional equation of L-function and a mean value theorem on the fourth moment of L-function are used.^a) In this paper, a new simplified analytic proof of the three primes theorem will be given which is not based on the Vinogradov’s estimation and the density theorem of zeros of L-function, and only some well-known simple facts on L-function are used…