Seminar L-functions

In the seminar we will study Dirichlet L-functions, which generalize the classical Riemann zeta function. We discuss their basic properties, such as the analytic continuation and the functional equation, and the rationality of some of their special values. Moreover, we investigate the connection of Dirichlet L-functions with the Dedekind zeta functions of quadratic number fields. As main applications, we prove Dirichlet's class number formula for quadratic number fields and Dirichlet's Theorem on arithmetic progressions.

Some familiarity with the basic notions of algebra (groups, rings, fields), complex analysis (holomorphic/meromorphic functions, the residue theorem) and elementary number theory (congruences, Legendre symbol, quadratic reciprocity) will be helpful.

General informations

The seminar takes place Tuesdays from 12-14 in HG F 26.5, starting on 05.10. until 21.12. (12 talks).

Two students share a talk. The talks should take about 90-120 minutes. You can do a board talk or use slides (e.g. Beamer LaTex). A script in Latex is required.

We follow the book of Don Zagier "Zetafunktionen und quadratische Körper"

Topics

Here is a list of topics.

The recordings of the talks can be found on polybox.

1. Dirichlet series Handout Slides part 1

2. The Riemann zeta function Handout part 1 Slides part 1 Handout part 2 Slides part 2

3. Dirichlet L-series Handout Slides part 1 Slides part 2

4. Special values of Dirichlet L-series Handout

5. Binary quadratic forms Handout

6. Class number formulas Handout

7. Quadratic forms and quadratic fields Handout

8. The zeta function of a quadratic field Handout

9. Genus theory Handout

10. Reduction theory

11. Values of zeta functions at s = 0 Handout

12. The Birch and Swinnerton-Dyer conjecture Handout part 1 Handout part 2