Seminar Lfunctions
In the seminar we will study Dirichlet Lfunctions, 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 Lfunctions 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 1214 in HG F 26.5, starting on 05.10. until 21.12. (12 talks).
Two students share a talk. The talks should take about 90120 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.

Dirichlet series Handout Slides part 1

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

Dirichlet Lseries Handout Slides part 1 Slides part 2

Special values of Dirichlet Lseries Handout

Binary quadratic forms Handout

Class number formulas Handout

Quadratic forms and quadratic fields Handout

The zeta function of a quadratic field Handout

Genus theory Handout

Reduction theory

Values of zeta functions at s = 0 Handout

The Birch and SwinnertonDyer conjecture Handout part 1 Handout part 2