Nardelli, on the physical interpretation of the riemann zeta function, the rigid surface operators in gauge theory, the adeles and ideles groups applied to various formulae regarding the riemann zeta function and the selberg trace formula, padic strings, zeta strings and padic cosmology and mathematical connections with some sectors of string theory and number theory preprint 102008. Books about analytic number theory in general that include some material about the riemann hypothesis or riemanns zeta function. Some transposition of those results to the riemann case can then be. If youre interested in eulers role in the development, i would look to dunhams euler. A generalized riemann zeta function known as the hurwitz zeta function can also be defined such that 14 the riemann zeta function may be computed analytically for even using either contour integration or parsevals theorem with the appropriate fourier series. Finally the general selberg trace formula is deduced easily from the properties of the selberg zetafunction. Finally the general selberg trace formula is deduced easily from the properties of the selberg zeta function. The precursors to the zeta function and its development by euler and riemann especially eulers original proof to the basel problem are fantastic. What are zeta functions of graphs and what are they good for. The authors discussion of the selberg trace formula stresses the analogy with the riemann zeta function. The eichlerselberg trace formula for levelone hecke. Many of the results about s that can be found with the usual tools of complex analysis have very important consequences on number theory see 4, p. 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. The fit to the bell curve is not yet good, but the riemann zeta function approaches its asymptotic behavior very slowly.
More generally, weil 9 noticed that the methods used to prove iii can be utilised to show the. Suggestions for exercises and further reading sec tion 15. Pdf the thermodynaic fomalism approach to selbergs zeta. Besides the classical approach to selbergs zeta function for cofi nite fuchsian groups s through the trace formula v there has been developed recently another one based on the thermodynamic. It is known that the function has infinitely many zeroes on this line in the complex plane. The riemann hypothesis is an unproven statement referring to the zeros of the riemann zeta function. The master of us all, and for riemann, one should turn to edwads riemanns zeta function to read riemanns. The contribution from the trivial zeros contains the barnes gfunction, while the other zeros generate the fredholm determinant of the laplacian.
The selberg zeta function is decomposed into a canonical product over its zeros in analogy with the hadamard product formula for the riemann zeta function. While selberg s zeta function resembles riemann s zeta in some. A conceptually easy corollary is the functional relation for the hurwitz zeta function. The selberg trace formula and the riemann zeta function. The prime number theorem, hardys theorem on the zeros of. An introduction jens marklof school of mathematics, university of bristol, bristol bs8 1tw, u. Differential geometry, selberg trace formula and riemann. Analytic continuation of the riemann zeta function 9. 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 relevance to the spectral interpretation of the riemann. Riemann zeta function and a geometric framework to which one can.
The fact that this function is analytic in this region of the complex plane is a consequence of the following basic fact. An approach to the selberg trace formula via the selberg. The zeta function is an important function in mathematics. 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. Pdf euler products of selberg zeta functions in the. Path integrals, hyperbolic spaces and selberg trace formulae. 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. Pdf the selberg trace formula bachelors degree thesis. T1 the selberg trace formula and the riemann zeta function. Number theory, trace formulas and discrete groups 1st. Motivated by the resemblance of the selberg trace formula to weils formula for the sum of zeros of the riemann zeta function, defined for any compact hyperbolic riemann surface a zeta functionlike expression, the selberg zeta function of a riemann surface.
There are various methods to derive this functional equation, see e. However, the formula 2 cannot be applied anymore if the real part. Path integrals, hyperbolic spaces and selberg trace. Statistical properties of zeta functions zeros arxiv. Bernhard riemann calculated the first six nontrivial zeros of the function and observed that they were all on the same straight line. Introduction to the selberg trace formula this is a talk about the paper h. Books that consist of collections of mathematical tables. Selbergs trace for mula as applied to a compact riemann surface 1972. Introduction we present a short and motivated proof of riemanns functional equation for riemanns zeta function s. Trace formulas, spectral geometry, lenght spectrum, geodesic flow. Wikipedia, selberg zeta function matthew watkins, citation collection on selberg trace formula and zeta functions. The notes give a direct approach to the selberg zetafunction for cofinite. 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. An approach to the selberg trace formula via the selberg zeta.
Chapter 9 the functional equation for the riemann zeta function. Besides the classical approach to selberg s zeta function for cofi nite fuchsian groups s through the trace formula v there has been developed recently another one based on the thermodynamic. Riemann zeta function we will eventually deduce a functional equation, relating s to 1 s. Quantum chaos, random matrix theory, and the riemann function. Johan andersson summation formulae and zeta functions. Selberg s zeta function is constructed in such a way that it has the same relation to selberg s trace formula as riemann s zeta function has to the socalled explicit formula for sums over riemann s zeros.
In mathematics, the selberg conjecture, named after atle selberg, is a theorem about the density of zeros of the riemann zeta function. Remarks on the pattersonselberg zeta function, black hole. This is known to be true for the first roots brent et al. We give a proof based on a functional equation for the jacobi theta function z p 1 m1 e. This paper explains the steps involved in the proof of the eichlerselberg trace formula for hecke operators of level one. While selbergs zeta function resembles riemanns zeta in some. The riemann zetafunction s is a meromorphic function. The selberg trace formula for psl 2,r volume 1 dennis a. Elementary spectral theory of automorphic functions 3. As can be seen, the first few nontrivial zeros occur at, 21. The master of us all, and for riemann, one should turn to edwads riemann s zeta function to read riemann s. Then i shall consider the kuznetsov trace formula followed by higher rank generalizations.
In a report published in 1859, riemann stated that this might very well be a general fact. Refinement of the spectral theory of automorphic functions 6. We shall then prove the positivity of the weil distribution assuming the validity of the analogue of the selberg trace formula. The riemann zeta function is defined as the analytic continuation of the function defined for. The authors discussion of the selberg trace formula stresses the analogy with the riemann zetafunction. 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. And, in a second time, we present selberg formula and its appli. Euler products of selberg zeta functions in the critical strip. Trace formula in noncommutative geometry and the zeros of the. Elliptic and parabolic conjugacy classes, and selberg super zeta functions grosche, c. 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.
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 hadamard factorization of the selberg zeta function on a. Extreme negative values near the riemann zeros are extremely scarce, as are large positive values. The second part contains lectures on selberg s 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. Chapter 9 the functional equation for the riemann zeta. By definition riemanns zeta function is given by the series. For simplicity, assume that mis a compact riemannian manifold. 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 selberg trace formula for the picard group sl2, zi. Selberg super trace formula for super riemann surfaces. 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 divisor of selberg s zeta function for kleinian groups patterson, s. The hadamard factorization of the selberg zeta function on. Titchmarsh, the theory of the riemann zeta function. The contribution from the trivial zeros contains the barnes g function, while the other zeros generate the fredholm determinant of the laplacian. Moreover, in 1859 riemann gave a formula for a unique the socalled holomorphic extension of the function onto the entire complex plane c except s 1. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number.
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. This course will be focused on trace formulae starting with the selberg trace formula for gl2 in the classical setting. In his lecture given in seattle in august 1996, on the occasion of the 100th anniversary of the proof of the prime number theorem, atle selberg comments about riemanns paper. An introduction to the theory of the riemann zetafunction. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. Notes on the riemann zeta function january 25, 2007 1 the zeta function 1. An introduction to selbergs trace formula department of. We see how the selberg trace formula and the eichlerselberg trace formula can be. In this chapter, we study the trace formula using the techniques of analytic number theory. Books that are paperlength say, under 50 pages doctoral dissertations published books based upon doctoral dissertations are, of course, included. References are elstrodt 8, hejhal 11, and terras 32. Trace formulas and zeta functions in spectral geometry.
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