Entry requirements

120 credits including 90 credits in mathematics with Complex Analysis and Real Analysis.

Learning outcomes

In order to pass 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.