Refinement of the spectral theory of automorphic functions 6. Differential geometry, selberg trace formula and riemann. The relevance to the spectral interpretation of the riemann. The trace formula for automorphic forms of weight m. For example, every automorphic function in if 29w has a fourier expansion with respect to the additive component of the adele group.
Throughout a career spanning more than six decades, professor selberg made significant contributions to modular forms, riemann and other zeta functions, analytic number theory. In this chapter, we study the trace formula using the techniques of analytic number theory. An approach to the selberg trace formula via the selberg. In mathematics, the poisson summation formula is an equation that relates the fourier series coefficients of the periodic summation of a function to values of the function s continuous fourier transform.
Special values of selbergs zetafunction sciencedirect. Zeta functions for the riemann zeros centre mersenne. While selberg s zeta function resembles riemann s zeta in some. Numerous and frequentlyupdated resource results are available from this search. Euler product, similar selberg trace and weil explicit formulas.
The contribution from the trivial zeros contains the barnes g function, while the other zeros generate the fredholm determinant of the laplacian. Apart from the basic spectral theory of the laplacian for cofinite groups the book is selfcontained and will be useful as a quick approach to the selberg zetafunction and the selberg. Zeros of the selberg zeta function for symmetric infinite. 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. Finally the general selberg trace formula is deduced easily from the properties of the selberg zetafunction. It describes the relationship between the geometric and spectral properties of such a group. Selberg supertrace formula for super riemann surfaces. It also bears a striking and mysterious resemblance to the selberg trace formula. The selberg trace formula for psl 2,r it seems that youre in usa.
The aim of this short lecture course is to develop selbergs trace formula for. Path integrals, hyperbolic spaces and selberg trace formulae. An approach to the selberg trace formula via the selberg zeta. 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. Kudryavtseva 1 filip saidak peter zvengrowski abstract an exposition is given, partly historical and partly mathematical, of the riemann zeta function s and the associated riemann hypothesis. Selberg super trace formula for super riemann surfaces. 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. Fischer, jurgen 1987, an approach to the selberg trace formula via the selberg zetafunction, lecture. The selberg trace formula is essentially equivalent to the statement that g c e as given by the gutzwiller trace formula, is equal to the trace ge given above for the motion of a particle on a compact surface of constant negative curvature. Suggestions for exercises and further reading sec tion 15. And conversely, the periodic summation of a function s. Book title the selberg trace formula for psl 2,r book subtitle volume 1 authors. The initial publication in 1956 of atle selberg dealt with this case, its laplacian differential operator and its powers. This chapter focuses on the selberg zetafunction of a kleinian group.
This is page contains resources dealing with the selberg trace formula and selberg zeta function. The riemann zetafunction universiteitsbibliotheek gent. Mckean, selbergs trace formula as applied to a compact riemann surface, comm. Motivated by the analogy, selberg introduced the selberg zeta function of a riemann surface, whose analytic properties are encoded by the selberg trace formula. Number theory, trace formulas and discrete groups sciencedirect. Kannan soundararajan selbergs contributions to the. 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. The interest of this case was the analogy between the formula obtained, and the. Number theory, trace formulas, and discrete groups. He was best known for his work on the riemann zeta function, on prime numbers and sieves, and on the spectral theory of automorphic forms. Selberg proved that the selberg zeta functions satisfy the analogue of the riemann hypothesis, with the imaginary parts of their zeros related to the eigenvalues of the laplacian operator of the riemann surface. His only paper on number theory was a report in 1859 to the berlin academy of sciences on \on the number of primes less than a given magnitude discussing s the riemann zeta function is the complex function s x1 n1 1 ns. Hejhal also informs me that selberg has obtained results similar to those of this paper, and that these will be appearing in hejhals forthcoming book on the trace formula. We describe two applications, including an interpretation of the selberg eigenvalue conjecture in terms of quadratic twists of certain dirichlet series, and a formula for an arithmetically weighted sum of the complete symmetric square lfunctions associated to cuspidal maass newforms of.
The divisor of selberg s zeta function for kleinian groups patterson, s. The riemann zeta function as a generating function in number theory chapter iii. Atle selberg 19172007 institute for advanced study. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. There are analogous functions arising in other elds such as selberg s zeta function of a riemann surface, iharas zeta function of a nite connected graph. Elliptic and parabolic conjugacy classes, and selberg super zeta functions grosche, c. 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. 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. In mathematics, the selberg conjecture, named after atle selberg, is a theorem about the density of zeros of the riemann zeta function. It is analogous to the famous riemann zeta function where is the set of prime numbers. Further support came in the 1970s and 80s from the works of mont. Zeta functions and chaos audrey terras october 12, 2009. How to understand the explicit formula for zeta function. T1 the selberg trace formula and the riemann zeta function.
We present several novel relations for selberg s zeta function for compact riemann surfaces. The riemann hypothesis for selbergs zetafunction and the. The results find direct application in calculating determ. Hejhal, the selberg trace formula for psl2,r volume i, p. The hadamard factorization of the selberg zeta function on. Using techniques similar to those of riemann, it is shown how to locate and count nontrivial zeros of s. The riemann hypothesis and the langlands program mathoverflow. The selberg trace formula arises by computing traced i kx, x dx, the integral of the kernel function on the diagonal. Let run over all distinct primitive ordered periodic geodesics, and let denote the positive length of, then the selberg zeta function is defined as. Apart from the basic spectral theory of the laplacian for cofinite groups the book is.
With the famous riemann hypothesis about the zeros of the riemann zeta function. We will consider the riemann hypothesis for the ihara zeta function and its connection with expander graphs. We give an account of a portion of the spectral theory. Edwards, 9780486417400, available at book depository with free delivery worldwide. The selberg zetafunction uses the lengths of simple closed geodesics instead of the primes numbers. These characters help to study the distribution of the primes in a given homology class. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. In the remaining part of this paper, we consider the avalues of the selberg zetafunction. By the selberg trace formula which we do not discuss here, there is a duality between the lengths of the primes and the spectrum of the laplace. If is a subgroup of sl2,r, the associated selberg zeta function is defined as follows. His paper on the selberg integral dates from this period and it is his only paper in norwegian.
This chapter provides an account of the special values of selberg s zetafunction. Spectral theory and the trace formula springerlink. Vinogradovs method in the theory of the riemann zeta function chapter v. Probably selberg s best and most important work is his trace formula for sl 2, which was done several years after the work for which he was awarded the fields medal. 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. We explore an idea of conrey and li of expressing the selberg trace formula as a dirichlet series. Quantum chaos, random matrix theory, and the riemann. The selberg trace formula is the analogue for these functions of the explicit formulas in prime number theory. 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. Eulers constants for the selberg and the dedekind zeta functions hashimoto, yasufumi, iijima, yasuyuki, kurokawa, nobushige, and wakayama, masato, bulletin of the belgian mathematical society simon stevin, 2004. Elementary spectral theory of automorphic functions 3. A formula that was originally developed in the context of number theory.
The selberg trace formula and the riemann zeta function. The selberg zetafunction was introduced by atle selberg. We obtain an asymptotic formula for the number of nontrivial avalues. The traces of powers of a laplacian can be used to define the selberg zeta function. As with the classical lfunctions, this function might be used to study the asymptotics of the prime geodesies. 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. Zeros of the derivatives of the riemann zetafunction ki, haseo and lee, yoonbok, functiones. Extreme negative values near the riemann zeros are extremely scarce, as are large positive values. Finally the general selberg trace formula is deduced easily from the properties of the selberg zeta function.
Wikipedia, selberg zeta function matthew watkins, citation collection on selberg trace formula and zeta functions. The definition and the simplest properties of the riemann zeta function chapter ii. We study the roots avalues of zs a, where zs is the selberg zeta function attached to a compact riemann surface. This zeta function is nowadays referred to as the selberg zeta function. In view of the above, it would be of interest to compare the avalue distribution for both zeta functions. Hejhal, the selberg trace formula and the riemann zeta function, duke. 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. 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. Grosche, c selberg supertrace formula for super riemann surfaces, analytic properties of the selberg super zeta functions and multiloop contributions to the fermionic string. Atle selberg 14 june 1917 6 august 2007 was a norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory. We will illustrate the method in the case of a compact riemann surface, and begin by recalling a few facts about the selberg zetafunction zs. This book provides an introduction to 1 various zeta functions for example, riemann, hurwitz, barnes, epstein, selberg, and ruelle, including graph zeta functions.
For simplicity, we will treat the trace formula only in the case of a compact quotient. Statistical properties of zeta functions zeros arxiv. 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 basic idea is to compute the trace of the iterated resolvent kernel of the hyperbolic laplacian in order to arrive at the logarithmic derivative of the selberg zeta function. We shall then prove the positivity of the weil distribution assuming the validity of the analogue of the selberg trace formula.
The product is over all primitive closed geodesics c in m d nh of poincare. Desy report desy 890120, february 1989, 1102, and commun. The selberg trace formula for modular correspondences. Motivated by the resemblance of the selberg trace formula to weils formula for the sum of zeros of the riemann zeta function, selberg 56 defined for any compact hyperbolic riemann surface a zeta functionlike expression, the selberg zeta function of a riemann surface. 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 riemann s explicit formula quite a bit. Moreover, the selberg zeta functions satisfy a riemann hypothesis trivially, since the zeroes correspond to the eigenvalues of an elliptic operator the trace formula relates these eigenvalues to lengths of closed curves on a riemann surface. The selberg trace formula for psl 2,r volume 1 dennis a. The selberg zetafunction was introduced by atle selberg 1956. Selberg used his trace formula to prove that the selberg zeta function of a riemann surface satisfies an analogue of the riemann hypothesis. The selberg trace formula for the picard group sl2, zi. The fit to the bell curve is not yet good, but the riemann zeta function approaches its asymptotic behavior very slowly.
Mathematical sciences research institute publications. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function s fourier transform. The selberg trace formula was introduced for cocompact discrete groups as an infinite analog of the kronecker classnumber formula. It is known that the function has infinitely many zeroes on this line in the complex plane. Selberg zeta functions over function fields request pdf. The notes give a direct approach to the selberg zetafunction for cofinite. By definition riemanns zeta function is given by the series. 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. On selbergs zeta function for compact riemann surfaces. It is analogous to the famous riemann zeta function. 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 selberg trace formula and the riemann zeta function, duke math. The selberg trace formula for psl 2,r book subtitle volume 1.
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