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Dynamical, Spectral, and Arithmetic Zeta Functions
Edited by: Michel L. Lapidus, University of California, Riverside, CA, and Machiel van Frankenhuysen, Rutgers University, Piscataway, NJ
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Contemporary Mathematics
2001; 195 pp; softcover
Volume: 290
ISBN-10: 0-8218-2079-6
ISBN-13: 978-0-8218-2079-7
List Price: US$57
Member Price: US$45.60
Order Code: CONM/290
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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

  • C.-H. Chang and D. H. Mayer -- Eigenfunctions of the transfer operators and the period functions for modular groups
  • C. Deninger and W. Singhof -- A note on dynamical trace formulas
  • C. E. Fan and J. Jorgenson -- Small eigenvalues and Hausdorff dimension of sequences of hyperbolic three-manifolds
  • A. Fel'shtyn -- Dynamical zeta functions and asymptotic expansions in Nielsen theory
  • W. F. Galway -- Computing the Riemann zeta function by numerical quadrature
  • S. Haran -- On Riemann's zeta function
  • M. L. Lapidus and M. van Frankenhuysen -- A prime orbit theorem for self-similar flows and Diophantine approximation
  • A. M. Odlyzko -- The \(10^{22}\)-nd zero of the Riemann zeta function
  • P. Perry -- Spectral theory, dymamics, and Selberg's zeta function for Kleinian groups
  • C. Soulé -- On zeroes of automorphic \(L\)-functions
  • H. M. Stark and A. A. Terras -- Artin \(L\)-functions of graph coverings
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