Zeta Functions and Chaos

A Window Into Zeta and Modular Physics MSRI Publications Volume 57, 2010 Zeta functions and chaos AUDREY TERRAS 1. Introduction This paper is an expanded version of lectures given at MSRI in June of 2008. It provides an introduction to various zeta functions emphasizing zeta functions of a finite graph and connections with random matrix theory and quantum chaos. For the number theorist, most zeta functions are multiplicative generating functions for something like primes (or prime ideals). The Riemann zeta is the chief example. There are analogous functions arising in other fields such as Selberg’s zeta function of a Riemann surface, Ihara’s zeta function of a finite connected graph. All of these are introduced in Section 2. We will consider the Riemann hypothesis for the Ihara zeta function and its connection with expander graphs. Chapter 3 starts with the Ruelle zeta function of a dynamical system, which will be shown to be a generalization of the Ihara zeta. A determinant formula is proved for the Ihara zeta function. Then we prove the graph prime number theorem. In Section 4 we define two more zeta functions associated to a finite graph: the edge and path zetas. Both are functions of several complex variables. Both are reciprocals of polynomials in several variables, thanks to determinant formulas. We show how to specialize the path zeta to the edge zeta and then the edge zeta to the original Ihara zeta. The Bass proof of Ihara’s determinant formula for the Ihara zeta function is given. The edge zeta allows one to consider graphs with weights on the edges. This is of interest for work on quantum graphs. See [Smilansky 2007] or [Horton et al. 2006b]. Lastly we consider what the poles of the Ihara zeta have to do with the eigenvalues of a random matrix. That is the sort of question considered in quantum chaos theory. Physicists have long studied spectra of Schrodinger¨ operators and random matrices thanks to the implications for quantum mechanics where eigenvalues are viewed as energy levels of a system. Number theorists such as A. 145 146 AUDREY TERRAS Odlyzko have found experimentally that (assuming the Riemann hypothesis) the high zeros of the Riemann zeta function on the line Re.s/ 1=2 have spacings D that behave like the eigenvalues of a random Hermitian matrix. Thanks to our two determinant formulas we will see that the Ihara zeta function, for example, has connections with spectra of more that one sort of matrix. References [Terras 2007] and [Terras 2010] may be helpful for more details on some of these matters. The first is some introductory lectures on quantum chaos given at Park City, Utah in 2002. The second is a draft of a book on zeta functions of graphs. 2. Three zeta functions 2.1. The Riemann zeta function Riemann’s zeta function for s C with 2 Re.s/ > 1 is defined to be Â Ã 1 X1 1 Y 1 .s/ 1 : D ns D ps n 1 p prime D In 1859 Riemann extended the definition of zeta to an analytic function in the whole complex plane except for a simple pole at s 1. He also showed that D there is a functional equation s Á .s/ s=2 .s/ .1 s/: (2-1) D 2 D The Riemann hypothesis, or RH, says that the nonreal zeros of .s/ (equiv- alently those with 0 < Re s < 1) are on the line Re s 1 . It is equivalent to D 2 giving an explicit error term in the prime number theorem stated below. The Riemann hypothesis has been checked to the 1013-th zero as of 12 October 2004, by Xavier Gourdon with the help of Patrick Demichel. See Ed Pegg Jr.’s website for an article called the Ten Trillion Zeta Zeros: http://www.maa.org/ editorial/mathgames. Proving (or disproving) the Riemann hypothesis is one of the million-dollar problems on the Clay Mathematics Institute website. There is a duality between the primes and the zeros of zeta, given analytically through the Hadamard product formula as various sorts of explicit formulas. See [Davenport 1980] and [Murty 2001]. Such results lead to the prime number theorem which says x # p prime p x ; as x : f D j g log x ! 1 The spacings of high zeros of zeta have been studied by A. Odlyzko; see the page www.dtc.umn.edu/~odlyzko/doc/zeta.htm. He has found that experimentally they look like the spacings of the eigenvalues of random Hermitian matrices (GUE). We will say more about this in the last section. See also [Conrey 2003]. ZETA FUNCTIONS AND CHAOS 147 EXERCISE 1. Use Mathematica to make a plot of the Riemann zeta function. Hint. The function Zeta[s] in Mathematica can be used to compute the Riemann zeta function. There are many other kinds of zeta function. One is the Dedekind zeta of an algebraic number field F such as Q.p2/ a bp2 a; b Q , where primes D f C j 2 g are replaced by prime ideals p in the ring of integers OF (which is ZŒp2 D a bp2 a; b Z , if F Q.p2/). Define the norm of an ideal of OF to be f C j 2 g D N a OF =a . Then the Dedekind zeta function is defined for Re s > 1 by D j j Y s 1 .s; F/ 1 N p ; D p where the product is over all prime ideals of OF . The Riemann zeta function is .s; Q/. Hecke gave the analytic continuation of the Dedekind zeta to all complex s except for a simple pole at s 1. And he found the functional equation D relating .s; F/ and .1 s; F/. The value at 0 involves the interesting number hF =the class number of OF which measures how far OF is from having unique factorization into prime numbers rather than prime ideals (h 1/. Also Q.p2/ D appearing in .0; F/ is the regulator which is a determinant of logarithms of 1 units (i.e., elements u OF such that u OF /. For F Q.p2/, the regulator 2 2 D is log1 p2. The formula is C hR .0; F/ ; (2-2) D w where w is the number of roots of unity in F (w 2 for F Q.p2/. One has D D .0; Q/ 1 . See [Stark 1992] for an introduction to this subject meant for D 2 physicists. 2.2. The Selberg zeta function. This zeta function is associated to a compact (or finite volume) Riemannian manifold. Assuming M has constant curvature 1, it can be realized as a quotient of the Poincare´ upper half-plane H x iy x; y R; y > 0 : D f C j 2 g The Poincare´ arc length element is dx2 dy2 ds2 C ; D y2 which can be shown invariant under fractional linear transformation az b z C ; where a; b; c; d R; ad bc > 0: ‘ cz d 2 C 148 AUDREY TERRAS It is not hard to see that geodesics — curves minimizing the Poincare´ arc length — are half-lines and semicircles in H orthogonal to the real axis. Calling these geodesics straight lines creates a model for non-Euclidean geometry since Euclid’s fifth postulate fails. There are infinitely many geodesics through a fixed point not meeting a given geodesic. The fundamental group of M acts as a discrete group of distance-pre- serving transformations. The favorite group of number theorists is the modular group SL.2; Z/ of 2 2 matrices of determinant one and integer entries or D the quotient = I . However the Riemann surface M SL.2; Z/ H D f˙ g D n is not compact, although it does have finite volume. Selberg defined primes in the compact Riemannian manifold M H D n to be primitive closed geodesics C in M . Here primitive means you only go around the curve once. Define the Selberg zeta function, for Re.s/ sufficiently large, as Y Y Z.s/ 1 e .s j/.C /: D C ŒC j 1 The product is over all primitive closed geodesics C in M H of Poincare´ D n length .C /. By the Selberg trace formula (which we do not discuss here), there is a duality between the lengths of the primes and the spectrum of the Laplace operator on M . Here Â @2 @2 Ã y2 : D @x2 C @y2 Moreover one can show that the Riemann hypothesis (suitably modified to fit the situation) can be proved for Selberg zeta functions of compact Riemann surfaces. EXERCISE 2. Show that Z.s 1/=Z.s/ has a product formula which is more C like that for the Riemann zeta function. The closed geodesics in M H correspond to geodesics in H itself. One D n can show that the endpoints of such geodesics in R (the real line = the boundary a b of H ) are fixed by hyperbolic elements of ; i.e., the matrices c d with trace a d > 2. Primitive closed geodesics correspond to hyperbolic elements that C generate their own centralizer in . Some references for this subject are [Selberg 1989] and [Terras 1985]. ZETA FUNCTIONS AND CHAOS 149 Figure 1. An example of a bad graph for zeta functions. 2.3. The Ihara zeta function. We will see that the Ihara zeta function of a graph has similar properties to the preceding zetas. A good reference for graph theory is [Biggs 1974]. First we must figure out what primes in graphs are.

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