Introduction

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0.1 From Riemann zeta function to dynamical

zeta functions

In this subsection we shall try to explain where dynamical zeta functions

come from . In a sense the study of dynamical zeta functions is part of the

theory of dynamical systems, but it is also intimately related to algebraic

geometry, number theory, topology and statistical mechanics.

0.1.1 Riemann zeta function

The theory of the Riemann zeta function and its generalisations represent

one of the most beatiful developments in mathematics. The Riemann zeta

function is that function defined on {s G W\ Re(s) 1} by the series

OO I

n=l

There is a second representation of ( which was discovered by Euler in 1749

and which is the reason for the significance of the Riemann zeta function in

arithmetic. This is Euler product formula:

as)= n

(i-p-)-1-

p prime

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Received by the editor June 22, 1998

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