Time:the second semester.

Credit:3 hours.

Period:54 hours.

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

Course contents:

(1) Bilinear Forms and the Large Sieve;

(2) Exponential Sums;

(3) Elementary Sieve Methods;

(4) The Dirichlet Polynomials;

(5) Zero Density Estimates;

(6) Character Sums;

(7) The critical zeros of the Riemann zeta function;

(8) The spacing of zeros of the Riemann zeta-function;

(9) Central values ofL-functions.

References:

(1) H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications 53, American Mathematical Society, 2004.

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

(3) Pan Chengdong and Pan Chengbiao, Fundamentals of analytic number

theory (in Chinese), Science Press. Beijing 1991.