We will consider the riemann hypothesis for the ihara zeta function and its connection with expander graphs. Statistical properties of zeta functions zeros arxiv. Riemann zeta function and a geometric framework to which one can. Zeros of the selberg zeta function for symmetric infinite. The fit to the bell curve is not yet good, but the riemann zeta function approaches its asymptotic behavior very slowly. However, the formula 2 cannot be applied anymore if the real part. Trace formulas, spectral geometry, lenght spectrum, geodesic flow. I hesitate to add to the chorus of praise here for h. This implies that the sum of the eigenvalues of the selfadjoint integral operator associated to the kernel function is the sum over the critical zeroes of the riemann zeta function. For example, correlations between the nontrivial zeros of the zeta function are believed to coincide asymptotically with those between the eigenvalues of large random unitary or hermitian matrices 7, 8, and the value distribution of arxiv. As with the classical lfunctions, this function might be used to study the asymptotics of the prime geodesies.
We shall then prove the positivity of the weil distribution assuming the validity of the analogue of the selberg trace formula. Path integrals, hyperbolic spaces and selberg trace formulae. The selberg trace formula for psl 2,r book subtitle volume 1. This zeta function is nowadays referred to as the selberg zeta function. More generally, weil 9 noticed that the methods used to prove iii can be utilised to show the.
And, in a second time, we present selberg formula and its appli. Further support came in the 1970s and 80s from the works of mont. Hyperbolic geometry and closed geodesics perhaps one of the easier routes into understanding dynamical zeta functions for ows is via. Spectral zeta functions of graphs and the riemann zeta function in the critical strip friedli, fabien and karlsson, anders, tohoku mathematical journal, 2017. The zeta function is defined in terms of the closed geodesics of the surface.
Trace formula in noncommutative geometry and the zeros of the. 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. T1 the selberg trace formula and 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. Refinement of the spectral theory of automorphic functions 6. Pdf cohomological theory of dynamical zeta functions. 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. Zeta functions and chaos audrey terras october 12, 2009. May 21, 2018 we discuss the riemann zeta function, the topology of its domain, and make an argument against. Suggestions for exercises and further reading sec tion 15. Differential geometry, selberg trace formula and riemann. In this chapter, we study the trace formula using the techniques of analytic number theory. For any hyperbolic surface of finite area there is an associated selberg zeta function.
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. I shall then explain that this leads to a natural spectral interpretation of the zeros of the riemann zeta function and a geometric framework in which the frobenius, its eigenvalues and the. The selberg trace formula for psl 2,r volume 1 dennis a. Trace formulas and zeta functions in spectral geometry. Moments of the riemann zeta function by kannan soundararajan in memoriam atle selberg abstract assuming the riemann hypothesis, we obtain an upper bound for the moments of the riemann zeta function on the critical line. The zeros and poles of the selberg zeta function, zs, can be described in terms of spectral data of the surface. 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. 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 zetafunction.
An approach to the selberg trace formula via the selberg zeta. The relevance to the spectral interpretation of the riemann. We derive this formula in the lemma and theorem below. This explicit formula involves the zeta function and, in particular, its nontrivial zeros.
By definition riemanns zeta function is given by the series. Hejhal, the selberg trace formula and the riemann zeta function, duke. Edwardss riemanns zeta function, for what little mathematics i have is self taught. Elementary spectral theory of automorphic functions 3. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number. Apart from the basic spectral theory of the laplacian for cofinite groups the book is. The fact that this function is analytic in this region of the complex plane is a consequence of the following basic fact. As a direct consequence we obtain a short, motivated proof of riemanns functional equation for s. Some transposition of those results to the riemann case can then be expected, in view of the formal analogies between the trace formulae for selberg zeta functions on the one hand, and weil explicit formula. The riemannvan mangoldt counting for mula states that n. Pdf polymeric quantum mechanics and the zeros of the.
Before this, we will state without proof, an essential theorem about the gamma function. Polymeric quantum mechanics and the zeros of the riemann zeta. Notes on the riemann zeta function january 25, 2007 1 the zeta function 1. The selberg trace formula for the picard group sl2, zi. Using techniques similar to those of riemann, it is shown how to locate and count nontrivial zeros of s. Riemann and his zeta function the university of north. The present book is concerned with these zeta functions from a cohomological point of view. The prime number theorem, hardys theorem on the zeros of. R, volume 1, springer lecture notes in mathematics no. These characters help to study the distribution of the primes in a given homology class. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. 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.
Selbergselberg trace formula trace formula on the universal covering tree. This chapter provides an account of the special values of selbergs zetafunction. Tooker occupy an obstinate refusal to conform, dunwoody, georgia, usa, 30338 dated. Introduction we present a short and motivated proof of riemanns functional equation for riemanns zeta function s. Originally, the selberg zeta function appeared in the spectral theory of automorphic forms and were suggested by an analogy between weils explicit formula for the riemann zeta function and selbergs trace formula 261. Zeros of the riemann zetafunction on the critical line. 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. The gamma function has an analytic continuation to. This conjecture is called the riemann hypothesis and is considered by many the greatest unsolved problem in mathematics. Finally the general selberg trace formula is deduced easily from the properties of the selberg zeta function. Edwards book riemanns zeta function 1 explains the historical context of riemanns paper, riemanns methods and results, and the. Riemann, dedekind, selberg, and ihara zetas audrey terras crm montreal 2009 more details can be found in my webpage.
Oct 06, 2016 polymeric quantum mechanics and the zeros of the riemann zeta function 10 e and the scale parameter 0, which can b e related to the. The selberg trace formula has been of great interest to mathematicians for almost 50 years. For a suitable kernel function on this space, the conjugacy class side of the selberg trace formula is the sum over the primes occurring in weils explicit formula. 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. The notes give a direct approach to the selberg zetafunction for cofinite. 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. 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. A conceptually easy corollary is the functional relation for the hurwitz zeta function. Nevertheless, after reading john derbyshires gripping prime obsession and following the math he used there with ease, i thought to tackle a more challenging book on the subject. Special values of selbergs zetafunction sciencedirect. Quantum chaos, random matrix theory, and the riemann 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. The character is given by the trace of certain functions on g the simplest case is when.
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. This is page contains resources dealing with the selberg trace formula and selberg zeta function. An approach to the selberg trace formula via the selberg. The aim of this short lecture course is to develop selbergs trace formula for.
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