A 11 Integers 11 B ( 2011 ) Dynamical Zeta Functions

These notes are a rather subjective account of the theory of dynamical zeta functions. They correspond to three lectures presented by the author at the “Numeration” meeting in Leiden in 2010. 1 A Selection of Zeta Functions In its various manifestations, a zeta function ζ(s) is usually a function of a complex variable s ∈ C. We will concentrate on three main types of zeta function, arising in three different fields, and try to emphasize the similarities and interactions between them. There are three basic questions which apply equally well to all such zeta functions: Question 1: Where is the zeta function defined and where does it have an analytic or meromorphic extension? Question 2: Where are the zeros of ζ(s)? What are the values of ζ(s) at particular values of s? Question 3: What does this tell us about counting quantities? We first consider the original and best zeta function. 1.1 The Riemann Zeta Function We begin with the most familiar example of a zeta function. INTEGERS: 11B (2011) 2 Number Theory Riemann ζ−function ζ−function Dynamical Systems Ruelle ζ−function Special values and volumes Mahler measures Surfaces with Surfaces with κ κ <0 =−1 QUE Class numbers of binary quadratic forms "Riemann Hypothesis" Selberg Geometry Figure 1: Three different areas and three different zeta functions Figure 2: Riemann (1826-1866): His only paper on number theory was a report in 1859 to the Berlin Academy of Sciences on “On the number of primes less than a given magnitude” discussing ζ(s) The Riemann zeta function is the complex function