Dynamical, Spectral, and Arithmetic Zeta Functions
About this Title
Michel L. Lapidus and Machiel van Frankenhuysen, Editors
Publication: Contemporary Mathematics
Publication Year : Volume 290
ISBNs: 978-0-8218-2079-7 (print); 978-0-8218-7880-4 (online)
MathSciNet review: 1868465
The original zeta function was studied by Riemann as part of his investigation of the distribution of prime numbers. Other sorts of zeta functions were defined for number-theoretic purposes, such as the study of primes in arithmetic progressions. This led to the development of $L$-functions, which now have several guises. It eventually became clear that the basic construction used for number-theoretic zeta functions can also be used in other settings, such as dynamics, geometry, and spectral theory, with remarkable results.
This volume grew out of the special session on dynamical, spectral, and arithmetic zeta functions held at the annual meeting of the American Mathematical Society in San Antonio, but also includes four articles that were invited to be part of the collection. The purpose of the meeting was to bring together leading researchers, to find links and analogies between their fields, and to explore new methods. The papers discuss dynamical systems, spectral geometry on hyperbolic manifolds, trace formulas in geometry and in arithmetic, as well as computational work on the Riemann zeta function.
Each article employs techniques of zeta functions. The book unifies the application of these techniques in spectral geometry, fractal geometry, and number theory. It is a comprehensive volume, offering up-to-date research. It should be useful to both graduate students and confirmed researchers.
Graduate students and research mathematicians interested in number theory.
Table of Contents
- Cheng-Hung Chang and Dieter H. Mayer – Eigenfunctions of the transfer operators and the period functions for modular groups [MR 1868466]
- Christopher Deninger and Wilhelm Singhof – A note on dynamical trace formulas [MR 1868467]
- Carol E. Fan and Jay Jorgenson – Small eigenvalues and Hausdorff dimension of sequences of hyperbolic three-manifolds [MR 1868468]
- Alexander Fel′shtyn – Dynamical zeta functions and asymptotic expansions in Nielsen theory [MR 1868469]
- William F. Galway – Computing the Riemann zeta function by numerical quadrature [MR 1868470]
- Shai Haran – On Riemann's zeta function [MR 1868471]
- Michel L. Lapidus and Machiel van Frankenhuysen – A prime orbit theorem for self-similar flows and Diophantine approximation [MR 1868472]
- A. M. Odlyzko – The -nd zero of the Riemann zeta function [MR 1868473]
- Peter Perry – Spectral theory, dynamics, and Selberg's zeta function for Kleinian groups [MR 1868474]
- C. Soulé – On zeroes of automorphic -functions [MR 1868475]
- H. M. Stark and A. A. Terras – Artin -functions of graph coverings [MR 1868476]