Graduate Seminar in Number Theory
Tel Aviv, Spring 2005
Zeev Rudnick
Schedule
Tueday 1416, Schreiber 8
The seminar will discuss topics in "analytic" number theory.
It will consist mostly of presentations by the students.
Possible Topics:

Arithmetic functions and function field arithmetic
 Arithmetic functions, Dirichlet convolution, Moebius inversion
 The Prime Number Theorem for F_{q}[x].

Binary quadratic forms
 binary quadratic forms, the modular group, reduction theory,
the class number.
 Unique factorization in imaginary quadratic fields.
 Dirichlet's class number formula.

Primes in arithmetic progression
 Dirichlet characters, orthogonality relations;
Dirichlet Lfunctions, analytic continuation to Re(s)>0.
 Infinite products and the Euler product for ζ(s)
and L(s, χ ).
 Primes in arithmetic progressions and the connection to L(1, χ ).

Sieve methods
 The sieve of Erathosthenes
 Selberg's upper bound sieve
 Applications: An upper bound for the number of twin primes up to x.

The zeros of Riemann's zeta function

The Gamma function and its properties, Stirling's formula.
 The Fourier transform and Poisson summation.
 The Riemann zeta function: analytic continuation and functional
equation.
 Entire functions of finite order and Weierstrass products
 The zeros of ζ(s) basics.
 The asymptotic formula for N(T),
the number of zeros up to height T.
 Relating zeros and primes: Riemann's explicit formula
 The prime number theorem  an overview
Prerequisites:
Knowledge of the contents of the following courses:

Introduction to Number Theory.
 complex variables
References
 H. Davenport, Multiplicative Number Theory
 T. Apostol, Introduction to Analytic Number Theory
 M. Ram Murty, Problems in Analytic Number Theory
 D. Cox, Primes of the form x^{2} + n y^{2}
 H. Cohn, Advanced Number Theory
Contact me at: rudnick@post.tau.ac.il
Office : Schreiber 316, tel: 6407806
Course homepage: http://www.math.tau.ac.il/~rudnick/courses/gradseminar2005.html