# Dynamical, Spectral, and Arithmetic Zeta Functions

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*Michel L. Lapidus; Machiel van Frankenhuysen*

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.

#### Readership

Graduate students and research mathematicians interested in number theory.

# Table of Contents

## Dynamical, Spectral, and Arithmetic Zeta Functions

- Contents vii8 free
- Preface ix10 free
- Eigenfunctions of the transfer operators and the period functions for modular groups 112 free
- A note on dynamical trace formulas 4152
- Small eigenvalues and Hausdorff dimension of sequences of hyperbolic three-manifolds 5768
- Dynamical zeta functions and asymptotic expansions in Nielsen theory 6778
- Computing the Riemann zeta function by numerical quadrature 8192
- On Riemann's zeta function 93104
- A prime orbit theorem for self-similar flows and Diophantine approximation 113124
- The 1022-nd zero of the Riemann zeta function 139150
- Spectral theory, dynamics, and Selberg's zeta function for Kleinian groups 145156
- On zeroes of automorphic L-functions 167178
- Artin L-functions of graph coverings 181192