RTG2491 Spring School
"Zeta functions, dynamics and analytic number theory”

The Mahler measure of a polynomial, defined as its geometric average over the unit torus, is a simple yet ubiquitous invariant, which appears in many different areas of mathematics. For example, it can be used as a measure of the complexity of the hypersurface cut out by the polynomial. It also allows one to compute the entropy of any action of a power of the additive group of the integers on a compact abelian group. Moreover, the Mahler measure of Alexander polynomials can be used to determine the asymptotic growth of homology in towers of knots, and similar results hold true for towers of finite graphs. Finally, Mahler measures of multivariate polynomials appear to be related to special values of L-functions. The aim of this minicourse is to provide an introduction to these (very) different areas in which Mahler measures make an appearance, with the hope of connecting the central themes of this workshop.
Lecture Notes

For a fixed compact hyperbolic manifold, we will naturally obtain two sequences of positive numbers. The first, known as the length spectrum, consists of the lengths of the manifold's closed geodesics. The second sequence comprises the eigenvalues of the Laplace operator's action on the manifold. In this course, we will investigate the Selberg trace formula, that is an identity relating these two sequences. We further study the so-called Selberg zeta function, that is encoding as an infinite product the length spectrum of the manifold similar to the Riemann zeta function encoding prime numbers.

The Riemann zeta function is a complex function that has become famous because of its connection with the distribution of prime numbers and the Riemann Hypothesis, a conjecture considered by many the most important unsolved problem in pure mathematics. The Riemann zeta function is a particular case of a Dirichlet series, which are series encoding arithmetic information of certain mathematical objects. In particular, one can define a Dirichlet series encoding information of groups. This allows one to recover group theoretical properties by investigating analytical properties of the Dirichlet series, in a similar way we can recover information about primes by looking at an analytical property of the Riemann zeta function. In this minicourse, Dirichlet series and zeta functions of groups will be introduced. We will cover basic properties, including abscissa of convergence, growth types and Euler decompositions.

In 1994 Motohashi proved in his ground-breaking work that the fourth moment of Riemann's Zeta function admits an interpretation as a third central moment of L functions summed over a basis of Maaß-cusp as well as holomorphic modular forms. The original proof uses sums of Kloosterman sums and the spectral interpretation arises via Kuznetsov's formula. In 2007, Bruggeman and Motohashi gave a much more direct proof that completely dispenses with Kloosterman sums. In this series of talks we will outline this new approach and obtain an intuitive understanding of why one should expect Motohashi's spectral expansion to have the shape it has.

Closest airports: Frankfurt Main International, Hannover

Göttingen is well connected via fast trains to most major cities in Germany. The travel time from Frankfurt is about 2h and from Hannover less than 1h.

Walking from the train station to the Mathematical Institut takes no more than 20 minutes passing through the town centre. Most hotels are in walking distance from the main building, however there is also a chance of using the buses.

Here is a list of recommended Hotels in Göttingen:

Leine Hotel
Eden Hotel 
Hotel Central
G Hotel
Novostar
Hotel Stadt Hannover

Here is a map where you can find bus stops, sightseeing points and other information about Göttingen. The link is set to the Autumn School room, you can move freely within the map.