Topics in multiplicative number theory

We will discuss asymptotic behaviour of expressions involving multiplicative functions. We start with the simplest topic on how to determine the average value of a multiplicative function, using both elementary and more powerful complex analytic tools, culminating in Halasz's theorem. Following up on that, we investigate the notorious Chowla conjecture which concerns self-correlations of the Möbius (and the closely related Liouville function) and which is a version of the heuristic that the multiplicative structures of consecutive integers are independent. We will discuss about some recent progress towards this conjecture. Topics that will be covered in the course: a) Mean values of multiplicative functions, Dirichlet series b) Distances of multiplicative functions, Wirsing's theorem and Halasz's theorem c) Multiplicative functions in short intervals: the theorem of Matomäki-Radziwill d) Applications of the Matomäki-Radziwill theorem: the logarithmic Chowla and Elliott conjectures for 2-point correlations. e) The local uniformity conjecture for the Liouville function e) Some connections with dynamical systems, Sarnak's conjecture and some applications of multiplicative number theory to Ramsey theoretic problems. The lecturer is not a leading expert and will be happy to explore the topics with the participants.