Advanced Analytic Number Theory

Time: the third semester

Course Classification: elective course

Credit: 3 hours

Period: 54 hours

Previous courses: Basic Analytic Number Theory

Syllabus:

(1) Elementary sieve methods

(2) Bilinear forms and the large sieve

(3) Exponential sums

(4) The Dirichlet polynomials

(5) Zero-density estimates

(6) Sums over primes

(7) Primes in arithmetic progressions

(8) The least prime in an arithmetic progression

(9) The Goldbach problem

(10) The circle method

References:

(1) H. Iwaniec and E. Kowalski, Analytic number theory,  AMS, 2004.

(2) E. Bombieri, Le grand crible dans la theorie analytique des nombres, societe mathematique de France, 1974.

(3) C. Pan and C. Pan, Fundamentals of analytic number theory (in Chinese), Science Press Beijing 1991.