Graduate Seminar in Number Theory

Tel Aviv, Spring 2005

Zeev Rudnick

Schedule

Tueday 14-16, 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

1. Arithmetic functions, Dirichlet convolution, Moebius inversion
2. The Prime Number Theorem for Fq[x].
• Binary quadratic forms

1. binary quadratic forms, the modular group, reduction theory, the class number.
2. Unique factorization in imaginary quadratic fields.
3. Dirichlet's class number formula.
• Primes in arithmetic progression

1. Dirichlet characters, orthogonality relations; Dirichlet L-functions, analytic continuation to Re(s)>0.
2. Infinite products and the Euler product for ζ(s) and L(s, χ ).
3. Primes in arithmetic progressions and the connection to L(1, χ ).
• Sieve methods

1. The sieve of Erathosthenes
2. Selberg's upper bound sieve
3. Applications: An upper bound for the number of twin primes up to x.
• The zeros of Riemann's zeta function

1. The Gamma function and its properties, Stirling's formula.
2. The Fourier transform and Poisson summation.
3. The Riemann zeta function: analytic continuation and functional equation.
4. Entire functions of finite order and Weierstrass products
5. The zeros of ζ(s)- basics.
6. The asymptotic formula for N(T), the number of zeros up to height T.
7. Relating zeros and primes: Riemann's explicit formula
8. The prime number theorem - an overview

Prerequisites:

Knowledge of the contents of the following courses:
1. Introduction to Number Theory.
2. 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 x2 + n y2
• H. Cohn, Advanced Number Theory

Contact me at: rudnick@post.tau.ac.il

Office : Schreiber 316, tel: 640-7806