Entry requirements

120 credit points with 90 credit points Mathematics including Complex Analysis

Learning outcomes

In order to pass the course (grade 3) the student should

know the definition of Riemann's zeta function and be familiar with its most important properties;

know the definition of Dirichlet characters and Dirichlet's L-function and be familiar with its most important properties;

know how to use the methods in the proof of the prime number theorem such as partial summation and integration, the Mellin transform and the inverse Mellin transform, and simple Tauberian theorems;

know the Dirichlet class number formula;

know Siegel's theorem and its proof;

be familiar with results on bounds of sums of characters;

know some sieve method and Bombieri's theorem.

Content

Results concerning the distribution of primes obtained by elementary methods. Dirichlet characters. The zeta function and Dirichlet's L-function. A proof of the prime number theorem and the prime number theorem for arithmetic sequences. Explicit formulas for Chebychev's psi-function. Dirichlet's class number formula. Siegel's theorem. Bounds on sums of characters. Briefly about sieve methods and Bombieri's theorem.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.