Time: the third Semester.
Credit: 3 hours.
Period: 54 hours.
Previous courses: Elementary Number Theory, Complex Analysis, Basic Analytic Number Theory
(1) The Riemann zeta-function and Dirichlet L-functions;
(2) Sums of two squares, and of two squares of primes;
(3) Exponential sums over primes: elementary method;
(4) The circle method in the Warning-Goldbach problem;
(5) The large sieve;
(6) The major arcs, Ⅰ;
(7) Exponential sums over primes: analytic mothod, Ⅰ;
(8) Sums of squares of primes;
(9) The major arcs, Ⅱ;
(10) Introduction to sieve methods;
(11) Exponential sums over primes: analytic mothod, Ⅱ;
(12) Mean-value theorems for exponential sums;
(13) Sums of cubes of primes;
(14) Sums of forth powers of primes;
(15) Sums of fifth and of seventh powers of primes.
(1) J. Y. Liu and T. Zhan, New development in the Additive theory of Prime Numbers, World Scientific Press, 2011
(2) L. K. Hua, Additive Theory of Prime Numbers