The Odd Goldbach Problem In Arithmetic Progressions

In this thesis we consider the solubility problem of linear equations, which is one important problem in classical analytic number theory. We extend Goldbach-Vinogradov's Theorem into arithmetic progressions, our result is as follows. Let k1, k2, k3 be any positive integers and l1,l2, l3 be integers satisfying (lj,kj) = 1, 1 < j < 3, and let N be a sufficiently large odd number satisfying N = l1 + l2 + l3(mod (k1, k2, k3)), and (li + lj - N, ki, kj) - 1, 1 ≤ i < j ≤ 3. Then there exists an effective computable constant 0 < δ < 1 such that K < Nδ, the equationN = p1 + p2 + p3, Pj = lj(mod kj), j = 1, 2, 3is solvable in primes p1, p2, p3, where K = max{2, k1, k2, k3}.Our result consists two important classical conclusions in analytic number theory. The first is I. M. Vinogradov' s three primes theorem that every sufficiently large odd number can be represented as the sum of three prime numbers. The second is Yu. V. Linnik's result on the bound estimate for the least prime in arithmetic progressions, i.e. there exsists an absolute constant c such that p(k,1) (?) kc, p = l+kn, n = 1, 2, ... In fact, we can get the former if k1 = k2 = k3 = 1 and the latter if k1, k2, k3 > 1 in our theorem.We prove our result by using Hardy-Littlewood's circle method. We apply I. M. Vinogradov's estimate for linear trigonometric sums over prime variables in arithmetic progressions to dispose the integration on the minor interval. For disposing the integration on the major interval, we apply the explicit results on prime numbers distribution, general Gauss sum, as well as estimate for the density of the zeros of Dirichlet L-functions.