Introduction it is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of. Eta invariant and selberg zeta function of odd type over convex cocompact hyperbolic manifolds. Conference zeta functions in geometry auguest 1418, 1990 tokyo institute of technology, tokyo, japan. It is not obvious how to adapt the invariance theory arguments of 3.
The series defining the eta and zeta functions are absolutely convergent for. Eta invariant and selberg zeta function of odd type over convex cocompact hyperbolic manifolds discussion of relation to l functions includes michael atiyah, h. The problem is analyzed from the point of view of geometry. It can be used to recover the etainvariant of operators in the family. Adiabatic limits of eta and zeta functions of elliptic. We also show that the selberg zeta function of odd type has a meromorphic. Jensen polynomials for the riemann zeta function and other. The special value is then called the eta invariant of the operator a. However, the formula 2 cannot be applied anymore if the real part. Both dirichlet eta function and riemann zeta function are special cases of polylogarithm. Basic zeta functions and some applications in physics.
Cheeger in, where they introduced the etaform of a family of elliptic operators as above. Pdf this paper concerns diractype operatorsp on manifoldsx with boundary. While the dirichlet series expansion for the eta function is convergent only for any complex number s with real part 0, it is abel summable for any complex number. I figured that, perhaps if i knew how to calculate these values, i would be able to generalize the technique to be able to calculate the corresponding. In particular, in the setting of diractype operators we obtain a purely analytical proof of a well known result of bransongilkey bg, which was obtained by invoking riemannian invariant theory. The operators q for our zeta and eta functions arise as follows. The eta function of d is related to the zeta function of an elliptic differential operator. We show the regularity of eta and zeta functions at s0. Riemanns zeta function if s 1 is a real number, then the series s x n 1 1 ns converges. Zeta and eta functions for atiyahpatodisinger operators. This function plays a central role in the study of the theory of modular functions and its applications to other areas. For a realizationp b defined by a boundary operatorb of atiyahpatodisinger type, the. The zeta function is an important function in mathematics.
We show that the eta invariant and the selberg zeta function of odd type satisfy certain functional. What is the analytic continuation of the riemann zeta function. Eta invariant and selberg zeta function of odd type over convex cocompact hyperbolic manifolds colin guillarmoua. An important feature of the eta invariant is that it is not locally computable. Eta invariant and selberg zeta function of odd type over convex cocompact hyperbolic manifolds article in advances in mathematics 2255. The prime number theorem, hardys theorem on the zeros of. Transformation formulas for generalized dedekind eta functions let. Analytic continuation of the riemann zeta function 9. The attentive reader can realize that our etainvariant is, in some sense, a generalized version of the etainvariant associated with the circle and with a hermitian representation defined in 11. In section 5 we will show that z 2 rmanifolds having nontrivial eta function exist for every r. The operators qfor our zeta and eta functions arise as follows. Project muse eta invariants and regularized determinants. More precisely, he showed that the rh is equivalent to the.
The eta invariant for even dimensional pin, manifolds peter b. The riemannsiegel formula and large scale computations of the riemann zeta function by glendon ralph pugh b. Author links open overlay panel colin guillarmou a sergiu moroianu b c jinsung park d. This zeta function is an in nite product over prime ideals pin o k, the ring of algebraic integers of k. The riemann zeta function is defined as the analytic continuation of the function defined for.
The heat equation gives a local formula for the index of any elliptic complex. Gilkey mathematics department, university of oregon, eugene, oregon 97403 in an earlier paper 1 l we used the eta invariant of the tangential spin, complex to calculate the complex ktheory of odd dimensional spherical space forms. The eta function and eta invariant of z2rmanifolds. The analogue of the dedekind eta function for cy manifolds i by jamey bass at santa cruz, and andrey todorov1 at santa cruz and so. Noncommutative geometry and the riemann zeta function alain connes according to my rst teacher gustave choquet one does, by openly facing a well known unsolved problem, run the risk of. For the love of physics walter lewin may 16, 2011 duration. Eta invariant and selberg zeta function of odd type over. We extend the calculus of adiabatic pseudodifferential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator. The present paper is devoted to further extension of this subject. C, and which play the part of a generalised fredholm determinant for the transfer operator lon a. The eta invariant for even dimensional pin, manifolds.
This book treats the atiyahsinger index theorem using heat equation methods. Cameron franc special values of riemanns zeta function. This is in contrast to the case of the zeta function. Riemann zeta function, the eta function is analogous to dirichlet l functions. Singularities of the eta function of firstorder differential. In this case one expects to have a product formula in terms of local factors. 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.
Pdf the eta function and eta invariant of z2rmanifolds. The eta invariant was introduced in the famous paper of atiyah. Mathematical society of japan, 1992 450 pp dates publication date. Dynamical zeta functions or dynamical determinants are power series z, respectively dz, which are constructed from weighted periodic orbit data arising from a, say, discretetime dynamical system f. Central to our approach is an interpretation involving closed. The basic example is the wellknown factorization of the riemann zeta function. A window into zeta and modular physics msri publications volume 57, 2010 basic zeta functions and some applications in physics klaus kirsten 1. The analogue of the dedekind eta function for cy manifolds i. Assume that we know the eigenvalues f gwith multiplicity of an essentially selfadjoint usually rst order di erential operator d. This is in contrast to the case of the zeta function bs 03bbsj of a positive elliptic operator b.
This paper concerns diractype operatorsp on manifoldsx with boundary which are producttype near the boundary. Eta invariant and selberg zeta function of odd type over convex cocompact hyperbolic manifolds colin guillarmou, sergiu moroianu, and jinsung park abstract. See at eta invariant relation to zeta function for more on this. Pdf zeta and eta functions for atiyahpatodisinger operators. Zeta functions in algebraic geometry mircea mustat. The eta invariant for a closed manifold is introduced by atiyah patodi. Colin guillarmou jad, sergiu moroianu imar, jinsung park kias.
The eta invariant is an essential ingredient in the computation of the virtual. The analogue of the dedekind eta function for cy threefolds, 2003 pdf. By a global zeta function we mean an object that takes into account all primes. Ken richardson, introduction to the eta invariant pdf. The next two sections develop the mathematical theory of and its zeros starting with basic results such as the intermediate value theorem from real analysis and the argument princi. I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation.
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. Dynamical zeta functions mark pollicott contents 1 introduction 2. Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps. For the dirac operator on a riemann surfacecomplex curve the eta function was discussed in see at zeta function of a riemann surface for more on this case related concepts. Publication information advanced studies in pure mathematics, volume 21 tokyo, japan. Smith, the eta invariant for a class of elliptic boundary value problems. We also define an eta invariant for the odd signature operator and, under some conditions, we describe it on the schottky space of 3dimensional schottky. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. The special value is then called the eta invariant of the operator. Therefore, we can introduce the corresponding zeta and eta functions. We show meromorphic extension and give a complete description of the divisors of a selberg zeta function of odd type zo. We use invariance theory to identify the integrand of the index theorem for the four classical elliptic complexes with the invariants of the heat equation. An example is considered in detail and a brief study of the relevant wodzicki function.
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