Entry requirements

120 credits including 90 credits in mathematics with Complex Analysis and Real Analysis. Proficiency in English equivalent to the Swedish upper secondary course English 6.

Learning outcomes

On completion of the course, the student should be able to:

• define fundamental objects appearing in the course such as the Gamma function, Theta functions, the Riemann Zeta function, Dirichlet L-functions, Dirichlet characters, and describe the most important properties of these;
• use the methods from the proof of the Prime Number Theorem, such as summation by parts, integration by parts, the Mellin transform and its inverse, and simple Tauberian Theorems;
• give an account of deductions and proofs of important results in the course such as Dirichlet's Class Number Formula, Jacobi's Theorems on the representation of integers as sums of squares, and apply such results in relevant situations.

Content

Results on the distribution of primes obtained by elementary methods. Dirichlet characters. The Gamma, Theta and Zeta functions, and Dirichlet L-functions. A proof of the Prime Number Theorem and its generalisation to arithmetic sequences. The explicit formulas for Chebyshev's Psi function. Dirichlet's Class Number Formula. Representation of integers as sums of squares. Special values of the Zeta function. Orientation about sieve methods and Bombieri's Theorem concerning the distribution of primes in arithmetic progressions.

Instruction

Lectures and problem solving sessions.

Assessment

Written examination at the end of the course and assignments during the course.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.