This course is an introduction to applications of probabilistic methods within number theory. We will discuss a selection of the topics presented in [1], starting out from the Erdős-Kac theorem about the distribution of number of distinct prime factors of a typical integer of size about N.
Possible topics include the distribution of values of the Riemann Zeta function, Chebychev bias (which concerns the question whether there are there more primes p Ξ 3 (mod 4) than primes p Ξ 1 (mod 4)) as well as connections between exponential sums and random walks.
Course structure: Lectures, homework, possibly presentations by course participants.
This course is aimed at a general PhD level audience and should be accessible and of interest to doctoral students from different directions, including combinatorial, analytic as well as algebraic or geometric directions.
Probabilistic methods and heuristics play an increasingly important role in many areas of mathematics with striking applications in proofs of deterministic results.
This course aims at developing a basic tool box of probabilistic methods based on applications within number theory. The course will start out from Emmanuel Kowalski's recent book [1] on this subject, but may include ad-ditional material in form of current research papers.
A background in number theory is not required for this course as we will follow Kowalski's approach and keep the number theoretic input at a minimum while focussing on the probabilistic tools.
Intended learning outcome: To understand and be able to apply probabilistic techniques to analyse the asymptotic behaviour of arithmetically defined probability measures. In other words, to gain a basic tool box of probabilistic tools.
