Advanced Number Theory 0366.4875.01
Prof. Zeev Rudnick
Tel Aviv, Spring 2012
The aim of the course is to understand the zeros of the Riemann zeta
function and other L-functions. Along the way we will develop some of
their applications in number theory and discuss Random Matrix Theory.
Prerequisites
The course is aimed at MSc and Ph.D. students.
I will assume knowledge of the courses:
complex function theory 1,
Introduction to number theory, real variables/measure theory (in
particular the Fourier transform), and probability theory.
Tentative syllabus
-
Primes and the Riemann zeta function; function field analogues
-
Basic properties of the Riemann zeta function
-
The Prime Number Theorem
-
The zeros of zeta and the Explicit Formula
- The pair correlation function
-
Random Matrix Theory, GUE/GOE and the circular ensembles
- L-functions, their zeros and level statistics.
Bibliography
For the classical background material on the Riemann zeta function
etc., consult any of the books
-
H. Davenport, Multiplicative Number Theory
-
T. Apostol, Introduction to Analytic Number Theory.
-
Hugh L. Montgomery, R. C. Vaughan,
Multiplicative Number Theory I:
Classical Theory
-
J. Stopple, A Primer of Analytic Number Theory: From Pythagoras to Riemann
The subject of the statistics of the zeros is new and there are no
text books. A reading list of articles will be handed out in due course.
Some surveys:
Homework
There will be periodic homework assignments which are
mandatory .
Assignment
#1 ,
#2 ,
#3 ,
#4
due date: June 7, 2012
Schedule
Thursday 13-16, Dan David 204.
Contact me at: rudnick@post.tau.ac.il, Office : Schreiber 316, tel: 640-7806
Course homepage: http://www.math.tau.ac.il/~rudnick/courses/advnt2012.html