Additive Theory of Prime Numbers

Time: the second semester

Course Classification: elective course

Credit: 3 hours

Period: 54 hours

Previous courses: Elementary Number Theory, Complex Analysis, Basic Analytic Number Theory

Syllabus:

(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 method, I

(8) Sums of squares of primes

(9) The major arcs, I

(10) Introduction to sieve methods

(11) Exponential sums over primes: analytic method, 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 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, American Mathematical Society, 2009.

(3) R. C. Vaughan, The Hardy-Littlewood Method, Cambridge University Press.