Time: the third Semester.
Credit: 3 hours.
Period: 54 hours.
Previous courses: Elementary Number Theory, Complex Analysis, Basic Analytic Number Theory
Course contents:
(1) The Riemann zetafunction and Dirichlet Lfunctions;
(2) Sums of two squares, and of two squares of primes;
(3) Exponential sums over primes: elementary method;
(4) The circle method in the WarningGoldbach 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) Meanvalue 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.
References:
(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
