Zeros of the selberg zeta function for noncompact surfaces. Here a discussion on the zeros of the selberg zetafunction is in order. An approach to the selberg trace formula via the selberg zeta. These statistical connections have since been extended to many other l functions in the selberg class 45, over function elds 39.
The selberg zetafunction was introduced by atle selberg. 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. For example, the canonical factorization theorem involves an analogue of the euler constant. Trace formula in noncommutative geometry and the zeros of. The analogy between zeta functions and trace formulas goes at least to selberg, when he proved his famous trace formula for hyperbolic surfaces and the result turned out to resemble weils generalization of riemanns explicit formula quite a bit. The trace formula for zeros of the selberg zeta function for groups with.
Trace formulas and zeta functions in spectral geometry. The selberg trace formula for automorphic forms of weightm, on bordered riemann surfaces is developed. How to understand the explicit formula for zeta function. 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. 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.
The zeta function and its relation to the prime number theorem ben rifferreinert abstract. 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. 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. Convergence of the normalized spectral counting function. We prove that the second integral of the spectral counting function has an asymptotic expansion out too. Differential geometry, selberg trace formula and riemann. This paper explains the steps involved in the proof of the eichlerselberg trace formula for hecke operators of level one. Euler product, similar selberg trace and weil explicit formulas. There was little evidence for this until 1956, when the selberg trace formula appeared. Further contributions developing a super analogue of the classical selberg trace formula, the selberg supertrace formula, are presented. Trace formulas, spectral geometry, lenght spectrum, geodesic flow. Finally the general selberg trace formula is deduced easily from the properties of the selberg zetafunction. Wikipedia, selberg zeta function matthew watkins, citation collection on selberg trace formula and zeta functions.
Interestingly, selberg showed that the riemann zeta function wasnt needed in counting primes. While selbergs zeta function resembles riemanns zeta in some. 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. Finally the general selberg trace formula is deduced easily from the properties of the selberg zeta function. Importance of the zeta function the zeta function is a very important function in mathematics. The role of the lattice lis taken over by the conjugacy classes qof g. The divisor of selbergs zeta function for kleinian groups patterson, s. 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. Zeta functions for the riemann zeros by andre voros 1. In the case of compact bordered riemann surfaces we can explicitly evaluate determinants of maasslaplacians for.
The determinant was defined via the holomorphy at the origin of the spectral zeta function of minakshisundaram and pleijel. R, volume 1, springer lecture notes in mathematics no. Euler products of selberg zeta functions in the critical strip. 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. We see how the selberg trace formula and the eichlerselberg trace formula can be deduced. Hejhalthe selberg trace formula and the riemann zeta function.
The selberg trace formula gives a relation between the primes of mand the eigenvalues of the poincare laplacian. Hejhal, the selberg trace formula and the riemann zeta function, duke. On the prime geodesic theorem for noncompact riemann. 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. In view of the above, it would be of interest to compare the avalue distribution for both zetafunctions. Pdf the selberg trace formula bachelors degree thesis. The authors discussion of the selberg trace formula stresses the analogy with the riemann zeta function. On selbergs zeta function for compact riemann surfaces. In those cases, all the spectrum of the laplacians are discrete. The explicit formula of guinand and weil can be written in the following way. The character is given by the trace of certain functions on g the simplest case is when. It is analogous to the famous riemann zeta function.
The prime number theorem, hardys theorem on the zeros of. It was shown by selberg in 1956 that for such surfaces the zeta function z x has an analytic. May 01, 2020 the selberg trace formula and the riemann zeta function. We will consider the riemann hypothesis for the ihara zeta function and its connection with expander graphs. Tools for the purpose could also be borrowed from spectral theory trace formulae, etc. 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. And, in a second time, we present selberg formula and its appli. An approach to the selberg trace formula via the selberg. Selberg zeta function and relative analytic torsion for.
Trace formula in noncommutative geometry and the zeros of the. The riemann hypothesis for selbergs zetafunction and the. 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. 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. We shall model our discussion on the selberg trace formula, but it di ers.
Riemann zeta function and a geometric framework to which one can. 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. The lectures cover a wide range of quantum chaos applications and can serve as a nonformal introduction to mathematical methods of quantum chaos. Analytic properties and the functional equation for the corresponding selberg super. Riemann hypothesis, functional determinant, wkb semiclassical approximation, trace formula,boltes law, quantum chaos.
More generally, weil 9 noticed that the methods used to prove iii can be utilised to show the. 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. Elliptic and parabolic conjugacy classes, and selberg superzeta functions grosche, c. Riemann zeta function and selberg zeta function let be a riemann surface with constant negative curvature and the modular. References are elstrodt 8, hejhal 11, and terras 32. Matthew watkins, citation collection on selberg trace formula and zeta functions. 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.
One key reason for this is that the selberg zetafunction has a much higher density of zeros at large height. It is obvious from the above examples that a detailed knowledge of the selberg zeta function is most desirable. 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. On the distribution of the avalues of the selberg zeta. The trace formula is formulated for arbitrary fuchsian groups of the first kind with reflection symmetry which include hyperbolic, elliptic and parabolic conjugacy classes. Selberg type zeta functions for the hilbert modular group. An introduction to selbergs trace formula department of.
Selberg supertrace formula for super riemann surfaces. While it was not created by riemann, it is named after him. New points of view on the selberg zeta function don. The zeta function is an important function in mathematics. In this chapter, we study the trace formula using the techniques of analytic number theory. Dynamical zeta functions for anosov systems via microlocal analysis by dyatlovzworski xiaolong han contents 1. What are zeta functions of graphs and what are they good for. The selberg zetafunction uses the lengths of simple closed geodesics instead of the primes numbers.
This zeta function is nowadays referred to as the selberg zeta function. The selberg trace formula and the riemann zeta function. In 1874, mertens 57 studied partial euler products for the classical riemann zeta function. In the remaining part of this paper, we consider the avalues of the selberg zetafunction. We shall then prove the positivity of the weil distribution assuming the validity of the analogue of the selberg trace formula. In fact, riemann conjecture topped hardys famous wish list from the 1920s. 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. 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. The divisor of selberg s zeta function for kleinian groups patterson, s. 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. The selberg zetafunction z attached to a finite volume riemann surface satisfies the functional equation z s. Zeta functions for the riemann zeros centre mersenne. 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. Generalized krein formula, determinants and selberg zeta function in even dimension colin guillarmou abstract.
An introduction jens marklof school of mathematics, university of bristol, bristol bs8 1tw, u. Elliptic and parabolic conjugacy classes, and selberg super zeta functions grosche, c. Zeta functions and chaos audrey terras october 12, 2009. 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. The selberg trace formula has been of great interest to mathematicians for almost 50 years.
The authors discussion of the selberg trace formula stresses the analogy with the riemann zetafunction. Spectral functions, special functions and the selberg zeta function. 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. T1 the selberg trace formula and the riemann zeta function. David lyon the physics of the riemann zeta function. The classical selberg zeta function for the modular group.
Much of that part of the theory of the riemann zetafunction based on the riemann hypothesis carries over to. The riemann zeta function is defined as the analytic continuation of the function defined for. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number. Quantum chaos, random matrix theory, and the riemann. 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. The selberg trace formula is an extension of fourier analysis to noncommutative operators on a compact riemann surface. 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.
The selberg trace formula and the selberg zeta function. We give a geometric interpretation of the explicit formulas of number theory as a trace formula on the noncommutative space of. Spectral functions, special functions and the selberg zeta. The eichlerselberg trace formula for levelone hecke. This paper deals with the calculation of contributions arising from elliptic and parabolic conjugacy classes to the selberg supertrace formula for super riemann surfaces. We will consider the riemann hypothesis for the ihara zeta. Mckean, selbergs trace formula as applied to a compact riemann surface, comm. Selberg trace formula for bordered riemann surfaces. 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. 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. Johan andersson summation formulae and zeta functions. Selberg super trace formula for super riemann surfaces. On zeros of the derivative of the threedimensional selberg zeta function minamide, makoto, illinois journal of mathematics, 2008.
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