A representation is obtained for the zeta function of the additive divisor problem. Spectral zeta function of a sublaplacian on product sub. Res 12 in the sense that for every compact subset kin the complement. This relationship along with the functional equation for the theta function will provide insight on the riemann zeta function. The unifying feature to the invariants of this paper is the zeta function associated to the conformal laplacian. Th is zeta function will be the sub ject of the present paper. The prime number theorem, hardys theorem on the zeros of. 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. Furthermore, we will introduce some other zeta function which are closly related to the riemann zeta function. We introduce zeta functions weighted by bounded operators and show how, in recent work, we solved a problem concerning the noncommutative integral that has been open for 20 years. This zetafunction will be the subject of the present paper. The spectral zeta function bernd ammann june 4, 2015 abstract in this talk we introduce spectral zeta functions.
The concept was generalized in several directions, e. The matrix is in sln,z and very exciting leads to a zeta function which always satisfies a functional equation. The constant y comes about from the decomposition of f fy. Eugene speer, on the structure of analytic renormalization, comm. Riemann and his zeta function 3 logz lnz the natural logarithm of a complex number z. The riemann zetafunction s is a meromorphic function. The zeta function for circular graphs oliver knill abstract. We look at entire functions given as the zeta function p 0 s, where are the positive eigenvalues of the laplacian of the discrete circular graph cn. Is there a relationship between the zeta function of a. Let be a generalized laplacian with notation as above. This zeta function is an in nite product over prime ideals pin o k, the ring of algebraic integers of k.
A window into zeta and modular physics msri publications volume 57, 2010 basic zeta functions and some applications in physics klaus kirsten 1. Laplacians and spectral zeta functions of totally ordered. Quantum chaos, random matrix theory, and the riemann. For example, the zeta function associated to the laplacian on the circle is just a multiple of the riemann zeta function. In mathematics, the riemann hypothesis is a conjecture that the riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 2. What are zeta functions of graphs and what are they good for. In number theory, the dedekind zeta function is associated to an algebraic number eld k, such as qp 2, for example. Zeta functions in algebraic geometry mircea mustat. We give a proof based on a functional equation for the jacobi theta function z p 1 m1 e. In section 7 we focus on a sublaplace operator on threedimensional heisenberg manifolds. The basic example is the wellknown factorization of the riemann zeta function. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. The ordinary dirichlet laplacian d2 dx2 is a special case of the sturmliouville operator when 1 2 and hence, mis the lebesgue measure. Mizuno, the spectral zeta function of the unit nsphere and an integral treated by ramanujan, kyushu j.
Chapter 9 the functional equation for the riemann zeta. The zeta function of the additive divisor problem and. On the basis of this representation, the meromorphic continuability of. In particular, the asymptotic expansion is for kt and not zs. For instance, this allows for the computation of the determinant of the laplace and dirac operators on a riemannian manifold, using the minakshisundarampleijel zeta function.
Bottom row their respective normalized level spacings. The zeta function of the laplacian on certain fractals article pdf available in transactions of the american mathematical society 3602 september 2005 with 38 reads how we measure reads. Pdf the zeta function of the laplacian on certain fractals. The most popular of which for computing functional determinants is the zeta function regularization. This zeta function will be the subject of the present paper. Hyperfunctions and spectral zeta functions of laplacians. Zeta function determinant of the laplace operator on theddimensional ball. 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. And then the laplacian which we define with this right side up triangle is an operator of f. 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. In the case of the laplacian on a riemannian manifold, the meromorphic con. Two times, the notation zs of the zeta function is used in place of the heat kernel kt. References are elstrodt 8, hejhal 11, and terras 32. Note that the usual definition of integration doesnt apply to the dirac delta function in one dimension, because it requires that the function be realvalued or complexvalued, as appropriate.
Frobenius number, riemannroch structure, and zeta functions 2 the kernel of this homomorphism is the. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number. The zeta function is an important function in mathematics. The spectral zeta function of the laplace beltrami operator was already introduced by minakshisundaram and pleijel in 1948, see 6. Alain connes, matilde marcolli, a walk in the noncommutative garden arxiv. However, the formula 2 cannot be applied anymore if the real part. Lectures on zeta functions over finite fields daqing wan department of mathematics, university of california, irvine, ca926973875 email. Many consider it to be the most important unsolved problem in pure mathematics bombieri 2000. Hessian of the zeta function for the laplacian on forms hessian of the zeta function for the laplacian on forms okikiolu, kate. K of a category k is an appropriate formulation of selbergtype. Riemann zeta spectrum adjacency matrix mathematica experiment with random 53regular graph 2000 vertices. The zeta function of the laplacian on certain fractals core. And its defined as fx,y is equal to three plus cosx2 multiplied by siny2.
By a global zeta function we mean an object that takes into account all primes. In this case one expects to have a product formula in terms of local factors. However, we have another strategy to introduce a zeta function of s we recall that the selberg zeta function zxs of a locally symmetric space x has a determinant expression via the laplacian. Hessian of the zeta function for the laplacian on forms. We essentially use the wellknown heat kernel expressions of the sublaplacian on g due to beals, gaveau and greiner. About a bosonic sibling of the connection laplacian. So the function that i had there was defined, its a twovariable function. O connor, determinants of laplacians, the raysinger torsion on lens spaces and the riemann zeta function, j. Spectral zeta function of the sublaplacian on two step. Regularized determinants of the laplacian 4 here zs. It is of great interest in number theory because it implies results about the distribution of prime numbers. The counting matrix of a simplicial complex, local pdf.
Pdf zeta function determinant of the laplace operator on. Zeta determinant of the laplacian on the real projective. We show that the spectral zeta function can be decomposed into spectral zeta functions of certain elliptic differential operators which in particular gives a and b. There are various methods to derive this functional equation, see e. Otherwise, it is also possible to consider the quotient of two. Riemann zeta function we will eventually deduce a functional equation, relating s to 1 s. Eigenvalues of the laplacian on inhomogeneous membranes. As i study the minakshisundarampleijel zeta function, i see that it has a lot in common with the selberg zeta function, in terms of analogous objects in algebraic geometry both have intimate connections to artin. The riemann zeta function, the laplacian, and how to.
Basic zeta functions and some applications in physics. Wikipedia, functional determinant zeta function version. The zeta function of a mathematical operator is a function defined as. 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 detailed analysis of the corresponding spectral zeta function is given in section 6. Pdf this work raises and addresses a question about the behaviour of the variations of the spectral zeta function. And here, id like to go through the computation involved in that. In mathematics, the laplace operator or laplacian is a differential operator given by the divergence of the gradient of a function on euclidean space.
We characterise the poles, compute their residues, and give expressions for some special values of the zetafunction. In the correct first paragraph, it is explained that zs as an analytic continuation, without pole at s0 and so the derivative at 0 makes sense. Introduction it is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of. Here g is an arbitrary nilpotent lie group of step 2 and we assume the existence of a lattice l. The selberg zeta function of a riemannian surface has a lot more setup, although a definition can be found in definition 4. It was professor yumtong siu who pointed out to us that any interesting property of the zeta function associated to the sublaplacian and its relations. We study the heat kernel trace and the spectral zeta function of an intrinsic sublaplace operator. Titchmarsh, the theory of the riemann zeta function. In particular, there is a relationship between the riemann zeta function and the mellin transform of the theta function.
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