Riemann and his zeta function 5 the second form for this integral is called the mellin transform 70 of 1 ex. We will consider the riemann hypothesis for the ihara zeta function and its connection with expander graphs. It is known that the function has infinitely many zeroes on this line in the complex plane. The rh has been proved for this zeta which is a rational function of u q. In section 7 we turn to another theme, the relation of special zeta values. We 11referto for multiple gamma functionsand 6, 7 for. Riemanns thinking on symmetrizing the zeta functional equation. Generalized krein formula, determinants, and selberg zeta function in even dimension by colin guillarmou abstract. In this paper, we study the ruelle zeta function and the selberg zeta functions attached to the fundamental representations for real hyperbolic manifolds with cusps. Milnorselberg zeta functions and zeta regularizations.
The fact that this function is analytic in this region of the complex plane is a consequence of the following basic fact. Pdf the functional equations of the selberg and ruelle. The selberg zetafunction was introduced by atle selberg 1956. Spectral zeta functions of graphs and the riemann zeta. Quantum chaos, random matrix theory, and the riemann. The purpose of this paper is to study the analytic properties of the higher selberg zeta function z. In 1942 atle selberg announced his basic estimate of the zeros of riemanns zeta function in the proceedings of the royal norwegian academy of sciences and letters dknvs. For the circle rz we can calculate the theta function explicitly. Zagier in tills paper we will consider the functions ez, p obtained by setting the complex variable s in the eisenstein series ez, s equal to a zero of the riemann zeta function and will show that these functions satisfy a number. Zeta functions have generally been associated with problems of arithmetic or algebra and tend to have common features. Observe that this function extends meromorphically to all of c with rst order poles in the set iz. The integral 1 can also be viewed as a laplace transform.
For irregular graphs there is no known functional equation and it is di. Differences of the selberg trace formula title and selberg. These notes grew originally from a set of conferences given on march 2010 on the \problems of the millenium\ at the complutense university in. Hecke lseries and rankin selberg l functions of classical. Interestingly, selberg showed that the riemann zeta function wasnt needed in counting primes. Mathematica experiment with random 53 regular graph 2000 vertices. The prime number theorem, hardys theorem on the zeros of. Pdf we obtain the approximate functional equation for the rankin selberg zetafunction on the 12line. Selberg zeta functions and regularized determinants. To prove a certain property of s that has an important role in the proof of riemanns functional equation we will introduce poisson summation formula. Generalized krein formula, determinants, and selberg zeta.
Moreover, introducing a polyselberg zeta function z. Quantum chaos, random matrix theory, and the riemann function. Garunkstis and simenas 5 for the extended selberg class. July 9, 2010 basic rankinselberg university of minnesota. The analytic continuation, the functional equation and the determinant expression of this function via the laplacian on a riemann surface are obtained.
The function mt has a meromorphic continuation to all of c. Further, we investigate the connection between the ruelle zeta function at the central. Titchmarsh, the theory of the riemann zeta function. Zeros of high derivatives of the riemann zeta function binder, thomas, pauli, sebastian, and saidak, filip, rocky mountain journal of mathematics, 2015. The selberg and ruelle zeta functions are dynamical zeta functions, which can associatedwiththegeodesic. Other dirichlet series and zeta functions for local and global ground fields, see 11r42, 11r52. For the selberg zeta function, the analog of rh is true see 3, chap. Lecture notes analytic number theory mathematics mit. In fact, riemann conjecture topped hardys famous wish list from the 1920s. Ihara zeta function and spectrum of the cone over a semiregular bipartite graph.
The classical selberg zeta function for the modular group. Department of mathematics, columbia university, new york, ny 10027, usa. The zeta function is defined in terms of the closed geodesics of the surface. The functional equation selberg zeta function satis es the following functional equation see hejhal 4. Again, the model is provided by the riemann zeta function. Each of these includes a discussion of the rankin selberg convolution and zeta function. Spectral zeta functions of graphs and the riemann zeta function in the critical strip friedli, fabien and karlsson, anders, tohoku mathematical journal, 2017. The research process has not been linear, suffering moments of advance and moments of stagnation, and radical changes in the tactic used to solve the problem. We give a proof based on a functional equation for the jacobi theta function z p 1. For any hyperbolic surface of finite area there is an associated selberg zeta function. The functional equation shows that the riemann zeta function has zeros at. Zeta functions come up in a lot of area of mathematics. We prove also a determinant representation of the zeta functions, using the regularized determinant of certain twisted differential operators.
The functional equations of the selberg and ruelle zeta functions for nonunitary twists. Zeta functions in algebraic geometry mircea mustat. Top row distributions for eigenvalues of a on left and imaginary parts of the zeta poles on right s. In this case, one expects the zeta function to be a rational function, in a suitable variable. Selbergs zeta function of a riemann surface, iharas zeta function of a. Chapter 9 the functional equation for the riemann zeta. The selberg and the riemann zeta functions have many similar properties. The general proof of this relation can be found in the article on the riemann zeta function that is posted on the class web site. This is nothing but the euler product expression of z. On the distribution of the avalues of the selberg zeta. Top row distributions for eigenvalues of a on left and imaginary parts of the zeta. The functional equations of the selberg and ruelle zeta functions for nonunitary twists polyxeni spilioti abstract. As a direct consequence we obtain a short, motivated proof of riemanns functional equation for s. There are various methods to derive this functional equation, see e.
Notes on the riemann zeta function january 25, 2007 1 the zeta function 1. The selberg zetafunction z attached to a finite volume riemann surface satisfies the functional equation z s. Chapter 3 starts with the ruelle zeta function of a dynamical system, which. Zeros of the riemann zetafunction on the critical line. Basic rankinselberg july 9, 2010 be the usual zeta function s x n 1 1 ns along with its gamma factor. Among other things, selberg found that there is a zeta function which corresponds to his trace formula in the same way that the riemann zeta function corresponds to the riemannweil explicit formula. The theory selbergof zeta functions for locally symmetric spaces ofrank one is evolved by 4 gangolli compactcase and gangolliwarner 5 noncompact case. Sis an entire analytic function satisfying a functional equation and a form of the riemann hypothesis. The functional equation for the riemann zeta function. The zeta function is an important function in mathematics. Hyperbolic geometry and closed geodesics perhaps one of the easier routes into understanding dynamical zeta functions for ows is via. In sections 3 and 4 we will consider the multivariable zeta functions known as edge. For a class of even dimensional asymptotically hyperbolic ah manifolds, we develop a generalized birmankrein theory to study scattering asymptotics and, when the curvature is constant, to analyze the selberg zeta function. The aim of these lectures is to provide an intorduc tion to the theory of the riemann zetafunction for stu dents who might later want to do research on the subject.
The functional equations of the selberg and ruelle zeta functions for nonunitary twists article pdf available july 2015 with 21 reads how we measure reads. The zeta functions of ruelle and selberg for hyperbolic. Basic rankin selberg july 9, 2010 be the usual zeta function s x n 1 1 ns along with its gamma factor. This zeta function is nowadays referred to as the selberg zeta function. We discuss the circumstances around this result, and give selberg s own account quoting from an interview with him. The above theorem is the exact analogue of the classical theory of eichler, shimura and manin of period polynomials of modular forms, according to which one can. The riemann zeta function david jekel june 6, 20 in 1859, bernhard riemann published an eightpage paper, in which he estimated \the number of prime numbers less than a given magnitude using a certain meromorphic function on c.
Furthermore we will introduce the mellin transform and its inversion formula, since in the proof of riemanns functional equation there will be several integrals that are precisely mellin transforms. In this paper, i will demonstrate an important fact about the zeros of the zeta function, and how it relates to the prime number theorem. The zeros and poles of the selberg zeta function, zs, can be described in terms of spectral data of the surface. Hilbert and p olya put forward the idea that the zeros of the riemann zeta function may have a spectral origin. We study the solutions of heckes functional equation in the framework of the extended selberg class. The selberg trace formula and the riemann zeta function. Eisenstein series and the riemann zetafunction by d.
The aim of these lectures is to provide an intorduction to the theory of the riemann zeta function for students who might later want to do research on the subject. In mathematics, the selberg conjecture, named after atle selberg, is a theorem about the density of zeros of the riemann zeta function. Diophantine equations the simplest zeta functions from algebraic geometry are the hasseweil. What are zeta functions of graphs and what are they good for. They are useful in studying the statistical properties of the objects in question. We introduce a selberg type zeta function of two variables which interpolates several higher selberg zeta functions.
Derek newlands experiments graph analog of odlyzko experiments for riemann zeta spectrum adjacency matrix. On the real part of nontrivial zeros of riemann zeta function 3. Sare related to the eigenvalues of the laplacebeltrami operator. A functional determinant for the riemann xi function. Multiple gamma functionsalso appear in functional equation for these selberg zeta functions. Atle selberg on the zeros of the zetafunction of riemann. M is encoded in these zeros according to the functional equation of the selberg zeta function. The singularities of m are contained in the set ijlmj ijlmjj lsnj. It is analogous to the famous riemann zeta function where is the set of prime numbers. An important instrument in the proofs of our results will be the functional equation which is investigated in the next section. Nt 27 sep 2006 higher selberg zeta functions for congruencesubgroups tetsuya momotani graduateschoolofmathematics,kyushuuniversity 6101,hakozakifukuo.
The zeta function and its relation to the prime number theorem ben rifferreinert abstract. Differences of the selberg trace formula title and. A note on heckes functional equation and the selberg class. Here a discussion on the zeros of the selberg zetafunction is in order. Chapter 9 the functional equation for the riemann zeta function.
The following notes summarize five years of effort. Zeros of the extended selberg class zetafunctions and of. The selberg zetafunction uses the lengths of simple closed geodesics instead of the primes numbers. Mit opencourseware makes the materials used in the teaching of almost all of mits subjects available on the web, free of charge. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin. In garunkstis and tamosiunas 4 the levinson and montgomery result was generalized to the lerch zeta function. Here local means relative to a prime p in z, or in some ring of integers in a number eld. So under this analogy the selberg zeta function for hyperbolic 3manifolds as well as the artin lfunction for a number field both are like an infinite product over primes prime geodesics in one case, prime ideals in the other, see also at spec z as a 3dimensional space containing knots of determinants. The functional equations of the selberg and ruelle zeta. In section 3, we prove theorem 1, and section 4 is devoted to the proofs of theorems 2, 3, and 4. Moreover, the montgomeryodlyzko behavior has been proved for function. Chapter 9 the functional equation for the riemann zeta function we will eventually deduce a functional equation, relating s to 1 s. Hierarchy of the selberg zeta functions springerlink. For the epstein zeta function of an zzary positive definite quadratic form, z2 1 generalizations of the selberg chowla formula for the binary case are obtained.
We can write the theta function in the form mt tre t p m. Pdf we obtain the approximate functional equation for the rankinselberg zetafunction on the 12line. Hierarchy of the selberg zeta functions request pdf. Lastly, but most importantly for number theory, the holomorphic behavior of the zeta function was linked to the distribution of prime numbers. With more than 2,200 courses available, ocw is delivering on the promise of open sharing of knowledge. Find, read and cite all the research you need on researchgate.
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