Some topics in analytic number theory
Student No.:50
Time:Tue/Thu 15:10-17:00, 2013-12-24 ~ 2014-1-16
Instructor:Han Wu  [Eidgenössische Technische Hochschule Zürich]
Place:Conference Room 3, floor 2, Jin Chun Yuan West Building
Starting Date:2013-12-24
Ending Date:2014-1-16








In the second part (3 lectures), we discuss the evolution of L-functions, leading to those associated with automorphic representations, passing through the class field theory. This part will be of type of survey.

In the third part (2 lectures), we discuss (generalized) Reimann hypothesis and Lindelöf hypothesis, as well as their relations and some applications.




For the first part: complex analysis, discrete Fourier transform

For the second part: Galois theory

For the third part: complex analysis



For the first part:

1.       N.M.Katz: sommes exponentielles.

2.       潘承洞 : 解析数论基础.

3.       S.J.Patterson: An Introduction to the Theory of Riemann Zeta-Function.


For the second part:

1.       S.S.Gelbart: Automorphic Forms on Adele Groups.

2.       S.Lang: Algebraic Number Theory.

3.       D.Ramakrishnan & R.J.Valenza: Fourier Analysis on Number Fields.

4.       A.Weil: Basic Number Theory.


For the third part:

1.       D.A.Burgess: On character sums and L-series I,II,III.

2.     S.J.Patterson: An Introduction to the Theory of Riemann Zeta-Function.



The course will be composed of 3 parts.

In the first part (2 lectures), we discuss the basic problems and tools in analytic number theory issued from the prime number theorem. We emphasize the interplay of the analytic properties of L-functions and the arithmetic of related arithmetic functions, as well as the role played by bounding exponential sums.