The exponential sum \(\sum exp(2 \pi i tr(AX+X^{-1])/n) plays an important role in the theory of Siegel modular forms. The sum is oves symmetric invertible matrices modulo n. We give universal bounds that hold for all A. The crux of the matter is tha case when n is prime, when we also prove that are bounds are optimal by evaluating the sum directly when A is a scalar matrix.

The investigation of L^2-Laplace eigenvalues and eigenfunctions for hyperbolic surfaces of finite area is a classical and exciting topic at the intersection of number theory, harmonic analysis and mathematical physics. In stark contrast, for (geometrically finite) hyperbolic surfaces of infinite area, the discrete L^2-spectrum is finite. A natural replacement or generalization are the resonances of the considered hyperbolic surface, which are the poles of the meromorphically continued resolvent of the Laplacian (and hence, each spectral parameter of an L^2 eigenfunction is indeed a resonance). Resonances are encoded in the zeros of the Selberg zeta function for the hyperbolic surface.

Resonances play an important role in number theory and various other fields, and many fascinating results about them have already been found; the generalization of Selberg's 3/16-theorem by Bourgain, Gamburd and Sarnak is a well-known example. However, an enormous amount of the properties of such resonances, also some very elementary ones, is still undiscovered. A few years ago, by means of numerical experiments, Borthwick noticed for some classes of Schottky surfaces (hyperbolic surfaces of infinite area without cusps and conical singularities) that their sets of resonances exhibit unexcepted and nice patterns, which are not yet fully understood.

After a brief survey of some parts of this field, we will discuss an alternative numerical method, combining tools from dynamics, zeta functions, transfer operators and thermodynamic formalism, functional analysis and approximation theory. The emphasis of the presentation will be on motivation, heuristics and pictures.