Vinogradov's theorem and its generalization on primes in arithmetic progression

Abstract

In this thesis, we first study in details Vinogradov's elegant adaptation of the Hardy-Littlewood circle method in proving Vinogradov's theorem : every sufficiently large odd number is a sum of three prime numbers. After that, we generalize the Weak Goldbach's Conjecture to the quadratic fields : given a quadratic field K with discriminant d, there exists an integer a, such that every sufficient large odd number N congruent a mod d is a sum of norm of three prime ideals of K. We proceed on to prove this conjecture by proving Vinogadrov's theorem for prime in arithmetic progression, i.e. let x1; x2; x3 and y be integers such that 1 < y and (xi; y) = 1 for i = 1; 2; 3, then for all sufficiently large odd integer N congruent to x1 + x2 + x3 mod y, there exist primes pi congruent xi mod y for i = 1; 2; 3, such that N = p1 + p2 + p3.