Soit $M$ une surface riemannienne à courbure $-1$ et à bord compact totalement géodésique. Il lui est attachée une fonction zêta $Z_M$, généralisant la fonction que Selberg introduisit pour les surfaces d'aire finie. L'expression de cette fonction en termes d'invariants spectraux (valeurs propres et résonances) établit son prolongement méromorphe au plan complexe. La preuve combine deux types de formules de trace, à la Selberg et à la Birman-Krein, en s'appuyant sur l'analyse du prolongement méromophe des résolvantes de divers laplaciens et des résultats de type diffusion (stationnaire et non stationnaire) qui en résultent.

Cyclotomic function fields was introduced and used by D. Hayes [15] in 1974 to construct the maximal abelian extension of rational function fields $F_q(T)$. Since then many basic properties on cyclotomic fields have been researched by S. Galovich and M. Rosen [9, 10], D. Goss [13, 14], etc. In this paper we survey several results on class number and cyclotomic units of cyclotomic function fields.

We will make a preparation for defining the Selberg zeta function of $PSL(2, \mathbf{Z}[T])$, which is a discrete subgroup of the loop group $G$ of $PSL(2, \mathbf{C})$. Conjugacy classes of $PSL(2, \mathbf{Z}[T])$ will be classified and the definition of the norm of hyperbolic classes will be proposed.

This article gives a survey of various generalizations of Riemann’s $\zeta$-function, associated with operator spectra and which may be generically called spectral zeta functions. Areas of application include Riemannian geometry (the spectrum of the Laplacian) and quantum mechanics. We review one example of each class in concrete detail: the Laplacian on a compact surface of constant negative curvature, and the Schrödinger operator on the real line with a homogeneous potential $q^{2M}$ ($M$ a positive integer).

The purpose of this Note is to provide a survey of some of the recent experimental work aimed at computing eigenvalues of the Laplacian for a variety of Hecke triangle groups $\mathbf{G}(2 \cos \frac{\pi}{N}) \setminus H$.