Zeta Functions and Chaos Audrey Terras October 12, 2009 Abstract: The zeta functions of Riemann, Selberg and Ruelle are briefly introduced along with some others. The Ihara zeta function of a finite graph is our main topic. We consider two determinant formulas for the Ihara zeta, the Riemann hypothesis, and connections with random matrix theory and quantum chaos. 1 Introduction This paper is an expanded version of lectures given at M.S.R.I. 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. Section 2. Three Zeta Functions 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. We will consider the Riemann hypothesis for the Ihara zeta function and its connection with expander graphs. Section 3. Ruelle’s zeta function of a Dynamical System, A Determinant Formula, The Graph Prime Number Theorem. The first topic is the Ruelle zeta function 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. Section 4. Edge and Path Zeta Functions and their Determinant Formulas, Connections with Quantum Chaos. 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 [42] or Horton, Stark and Terras [23]. 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 Schrödinger 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. 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 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 [50] and [51] 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 Riemann’sZeta Function Riemann’szeta function for s C with Re(s) > 1 is defined to be 2 1 1 1 1 (s) = = 1 . ns ps n=1 p=prime X Y 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 there is a functional equation s/2 s (s) = ( )(s) = (1 s). (1) 2 The Riemann hypothesis (or RH) says that the non-real zeros of (s) (equivalently those with 0 < Re(s) < 1) are on 1 the line Re(s) = 2 . It is equivalent to giving an explicit error term in the prime number theorem stated below. The Riemann 1 hypothesis has been checked to 1013-th zero. (October 12th 2004), by Xavier Gourdon with the help of Patrick Demichel. See Ed Pegg Jr.’swebsite 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 1 million dollar problems on the Clay Math. 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 [11] and Murty [36]. Such results lead to the prime number theorem which says x # p = prime p x , as x . f j ≤ g log x ! 1 The spacings of high zeros of zeta have been studied by A. Odlyzko www.dtc.umn.edu/~odlyzko/doc/zeta.htm who 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 [10]. Exercise 1 Use Mathematica to do a plot of the Riemann zeta function. Hint. Mathematica has a command to give you the Riemann zeta function. It is Zeta[s]. 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 are replaced by prime ideals p in the ring of integers OF (which is Z[p2] = a + f j 2 g f bp2 a, b Z , if F = Q(p2)). Define the norm of an ideal of OF to be Na = OF /a . Then the Dedekind zeta functionj 2is definedg for Re s > 1 by j j s 1 (s, F ) = 1 Np , p Y 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 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 appearing in (0,F ) is the regulator which is a determinant of logarithms of units (i.e., elements u O Q(p2) F 1 2 such that u OF ). For F = Q(p2), the regulator is log 1 + p2 . The formula is 2 hR (0,F ) = , (2) w p 1 where w is the number of roots of unity in F (w = 2 for F = Q( 2). One has (0, Q) = 2 . See Stark [43] for an introduction to this subject meant for 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 Poincaré upper half plane H = x + iy x, y R, y > 0 . f j 2 g The Poincaré arc length element is dx2 + dy2 ds2 = y2 which can be shown invariant under fractional linear transformation az + b z , where a, b, c, d R, ad bc > 0. ! cz + d 2 2 It is not hard to see that geodesics which are curves minimizing the Poincaré 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’s5th 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-preserving 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 the quotient = / I . However the Riemann surface M = SL(2, Z) H is not compact, although it does have finite volume. Selberg definedf primesg in the compact Riemannian manifold M n= H to be primitive closed geodesics C in M. Here primitive means you only go around the curve once. n Define the Selberg zeta function, for Re(s) suffi ciently large, as (s+j)(C) Z(s) = 1 e . [C] j 1 Y Y≥ The product is over all primitive closed geodesics C in M = H of Poincaré length (C). By the Selberg trace formula (which we do not discuss here), there is a duality between the lengthsn of the primes and the spectrum of the Laplace operator on M. Here @2 @2 = y2 + . @x2 @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 like that for the Riemann zeta function. The closed geodesics in M = H correspond to geodesics in H itself. One can show that the endpoints of such geodesics n a b in (the real line = the boundary of H) are fixed by hyperbolic elements of ; i.e., the matrices with trace R c d a + d > 2. Primitive closed geodesics correspond to hyperbolic elements that generate their own centralizer in. Some references for this subject are Selberg [41] and Terras [48]. 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 [7]. First we must figure out what primes in graphs are. Recalling what they are for manifolds, we expect that we need to look at closed paths that minimize distance.