The riemann zeta function is defined as the analytic continuation of the function defined for. Quantum chaos, random matrix theory, and the riemann function. While selbergs zeta function resembles riemanns zeta in some. Elliptic and parabolic conjugacy classes, and selberg superzeta functions grosche, c. The selberg trace formula gives a relation between the primes of mand the eigenvalues of the poincare laplacian. These statistical connections have since been extended to many other l functions in the selberg class 45, over function elds 39. There are analogous functions arising in other elds such as selbergs zeta function of a riemann surface, iharas zeta function of a nite connected graph. There was little evidence for this until 1956, when the selberg trace formula appeared.
Hejhalthe selberg trace formula and the riemann zeta function. We give a geometric interpretation of the explicit formulas of number theory as a trace formula on the noncommutative space of. Hejhal, the selberg trace formula and the riemann zeta function, duke. Euler products of selberg zeta functions in the critical strip. The prime number theorem, hardys theorem on the zeros of. Tools for the purpose could also be borrowed from spectral theory trace formulae, etc. For example, the canonical factorization theorem involves an analogue of the euler constant. In those cases, all the spectrum of the laplacians are discrete.
Pdf the selberg trace formula bachelors degree thesis. Matthew watkins, citation collection on selberg trace formula and zeta functions. Euler product, similar selberg trace and weil explicit formulas. May 01, 2020 the selberg trace formula and the riemann zeta function. The determinant was defined via the holomorphy at the origin of the spectral zeta function of minakshisundaram and pleijel. Trace formula in noncommutative geometry and the zeros of. An approach to the selberg trace formula via the selberg zeta. Zeta functions for the riemann zeros by andre voros 1.
The role of the lattice lis taken over by the conjugacy classes qof g. Selberg type zeta functions for the hilbert modular group. In the case of compact bordered riemann surfaces we can explicitly evaluate determinants of maasslaplacians for. The lectures cover a wide range of quantum chaos applications and can serve as a nonformal introduction to mathematical methods of quantum chaos. Interestingly, selberg showed that the riemann zeta function wasnt needed in counting primes. The selberg zeta function was introduced by selberg as an analogue of the riemann zeta function where the prime numbers are replaced by the lengths. How to understand the explicit formula for zeta function. We shall then prove the positivity of the weil distribution assuming the validity of the analogue of the selberg trace formula. The selberg trace formula for automorphic forms of weightm, on bordered riemann surfaces is developed. Analytic properties and the functional equation for the corresponding selberg super. The selberg trace formula has been of great interest to mathematicians for almost 50 years. Here a discussion on the zeros of the selberg zetafunction is in order. Differential geometry, selberg trace formula and riemann. Generalized krein formula, determinants and selberg zeta function in even dimension colin guillarmou abstract.
The trace formula for zeros of the selberg zeta function for groups with. Dynamical zeta functions for anosov systems via microlocal analysis by dyatlovzworski xiaolong han contents 1. Request pdf differential geometry, selberg trace formula and riemann hypothesis an integral representation for the adjoint exterior derivative is introduced according to the foundations of. Motivated by analogy between this trace formula and the explicit formulas of number theory relating the zeroes of the riemann zeta function to prime numbers, selberg 1introduced a zeta function whose analytic properties are encoded in the selberg trace formula. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. We see how the selberg trace formula and the eichlerselberg trace formula can be deduced. We give a spectral interpretation of the critical zeros of the riemann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances. Convergence of the normalized spectral counting function on degenerating hyperbolic riemann surfaces of finite volume. In 1874, mertens 57 studied partial euler products for the classical riemann zeta function. 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 riemann weil explicit formula.
Spectral functions, special functions and the selberg zeta function. In fact, riemann conjecture topped hardys famous wish list from the 1920s. Atle selberg, harmonic analysis and discontinuous groups in weakly symmetric riemannian spaces with applications to dirichlet series, journal of the indian mathematical society 20 1956 4787. 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 selberg zetafunction was introduced by atle selberg. In view of the above, it would be of interest to compare the avalue distribution for both zetafunctions. In mathematics, the selberg trace formula, introduced by selberg 1956, is an expression for the character of the unitary representation of g on the space l 2 g. Theta and selberg zeta function ulrich bunke june 22, 2015 abstract contents 1 the theta function 1 2 hyperbolic space 5 3 the selberg trace formula 8 4 the selberg zeta function 10 5 a panorama of results on z s 11. It was shown by selberg in 1956 that for such surfaces the zeta function z x has an analytic.
Elliptic and parabolic conjugacy classes, and selberg super zeta functions grosche, c. The languages are analytic number theory and the theory of functions. Mckean, selbergs trace formula as applied to a compact riemann surface, comm. New points of view on the selberg zeta function don. Zeros of the selberg zeta function for noncompact surfaces. We shall model our discussion on the selberg trace formula, but it di ers. On the prime geodesic theorem for noncompact riemann. Selberg zeta function and relative analytic torsion for. In this case the selberg trace formula is formally similar to the explicit formulas relating the zeros of the riemann zeta function to prime numbers, with the zeta zeros corresponding to eigenvalues of the laplacian, and the primes corresponding to geodesics. The zeta function is an important function in mathematics. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number. In addition, an overview is presented on some recent achievements in the theory of the selberg trace formula on riemann surfaces, its super generalization, their use in mathematical physics and string theory, and some further results derived from the selberg super trace formula. In the paper kw2 we introduced a new type of selberg zeta function for establishing a certain identity among the nontrivial zeroes of the selberg zeta function and of the riemann zeta function. The authors discussion of the selberg trace formula stresses the analogy with the riemann zeta function.
An introduction jens marklof school of mathematics, university of bristol, bristol bs8 1tw, u. The selberg trace formula is an extension of fourier analysis to noncommutative operators on a compact riemann surface. The selberg trace formula and the riemann zeta function. Further contributions developing a super analogue of the classical selberg trace formula, the selberg supertrace formula, are presented. Trace formulas and zeta functions in spectral geometry. Importance of the zeta function the zeta function is a very important function in mathematics.
The authors discussion of the selberg trace formula stresses the analogy with the riemann zetafunction. The guinandweil explicit formula without entire function. On the prime geodesic theorem for noncompact riemann surfaces. The riemann hypothesis for selbergs zetafunction and the. Trace formula in noncommutative geometry and the zeros of the. We prove that the second integral of the spectral counting function has an asymptotic expansion out too. The second part contains lectures on selbergs considerable research studies on understanding the principles of several aspects of mathematics, including in modular forms, the riemann zeta function, analytic number theory, sieve methods, discrete groups, and trace formula. Finally the general selberg trace formula is deduced easily from the properties of the selberg zetafunction. It is obvious from the above examples that a detailed knowledge of the selberg zeta function is most desirable. We will consider the riemann hypothesis for the ihara zeta function and its connection with expander graphs. T1 the selberg trace formula and the riemann zeta function. On selbergs zeta function for compact riemann surfaces.
Much of that part of the theory of the riemann zetafunction based on the riemann hypothesis carries over to. Wikipedia, selberg zeta function matthew watkins, citation collection on selberg trace formula and zeta functions. David lyon the physics of the riemann zeta function. The explicit formula of guinand and weil can be written in the following way. Riemann zeta function and a geometric framework to which one can. Convergence of the normalized spectral counting function. Zeta functions for the riemann zeros centre mersenne. In this chapter, we study the trace formula using the techniques of analytic number theory. While it was not created by riemann, it is named after him. And, in a second time, we present selberg formula and its appli. Quantum chaos, random matrix theory, and the riemann. 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 selberg zeta function.
This closed orbit sum is nothing but a special case of selberg s trace formula 4 as applied to compact riemann surfaces, and the final result can again be formulated in terms of the selberg zeta function. The classical selberg zeta function for the modular group. This work proposes to investigate certain meromorphic functions defined by dirichlet series over the nontrivial zeros p of the riemann zeta function s, and to thoroughly compile their explicit analytical features. An approach to the selberg trace formula via the selberg. Selberg supertrace formula for super riemann surfaces.
In the remaining part of this paper, we consider the avalues of the selberg zetafunction. We will consider the riemann hypothesis for the ihara zeta. Hierarchy of the selberg zeta functions request pdf. R, volume 1, springer lecture notes in mathematics no. Selberg super trace formula for super riemann surfaces. Finally the general selberg trace formula is deduced easily from the properties of the selberg zeta function. It is analogous to the famous riemann zeta function. The trace formula is formulated for arbitrary fuchsian groups of the first kind with reflection symmetry which include hyperbolic, elliptic and parabolic conjugacy classes. Selbergs zeta function is constructed in such a way that it has the same relation to selbergs trace formula as riemanns zeta function has to the socalled explicit formula for sums over riemanns zeros. References are elstrodt 8, hejhal 11, and terras 32. 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. The selberg zetafunction uses the lengths of simple closed geodesics instead of the primes numbers.
What are zeta functions of graphs and what are they good for. The aim of this short lecture course is to develop selbergs trace formula for. Theta and selberg zeta function ulrich bunke june 22, 2015 abstract contents 1 the theta function 1 2 hyperbolic space 5 3 the selberg trace formula 8. This paper deals with the calculation of contributions arising from elliptic and parabolic conjugacy classes to the selberg supertrace formula for super riemann surfaces. The character is given by the trace of certain functions on g the simplest case is when. Riemann hypothesis, functional determinant, wkb semiclassical approximation, trace formula,boltes law, quantum chaos. The divisor of selberg s zeta function for kleinian groups patterson, s.
An introduction to selbergs trace formula department of. Yeah, im jealous the riemann hypothesis is named after the fact that it is a hypothesis, which, as we all know, is the largest of the three sides of a right triangle. Spectral functions, special functions and the selberg zeta. Selberg trace formula for bordered riemann surfaces. Trace formulas, spectral geometry, lenght spectrum, geodesic flow. This paper explains the steps involved in the proof of the eichlerselberg trace formula for hecke operators of level one. An approach to the selberg trace formula via the selberg zetafunction 1987 padic numbers, padic analysis, and zetafunctions 1984 the selberg trace formula for psl 2,ir vol. By focusing on the selberg zeta function, we can prove some deep results about the distribution of eigenvalues and pseudoprimes lengths of closed geodesics for a compact riemann surface.
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. On the distribution of the avalues of the selberg zeta. The eichlerselberg trace formula for levelone hecke. The volume also contains results on the study of the properties of a particular integrable billiard system in the hyperbolic plane, a proposal concerning interbasis expansions for spheroidal coordinate systems in fourdimensional euclidean space, and some further results derived from the selberg super. More generally, weil 9 noticed that the methods used to prove iii can be utilised to show the. The selberg trace formula of compact riemann surfaces 3 the selberg trace formula relates the trace trexp t x1 j1 exp t j of the heat kernel to the kind of dual theta function. Zeta functions and chaos audrey terras october 12, 2009. The selberg trace formula and the selberg zeta function. Selberg supertrace formula for super riemann surfaces ii. Atle selberg 14 june 1917 6 august 2007 was a norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. This zeta function is nowadays referred to as the selberg zeta function. Johan andersson summation formulae and zeta functions. On zeros of the derivative of the threedimensional selberg zeta function minamide, makoto, illinois journal of mathematics, 2008. The divisor of selbergs zeta function for kleinian groups patterson, s.