The relationships between the spectral zeta function, zeta kernel and heat kernel are used to aid the construction and discuss other spectral quantities e. The exposition is selfcontained, and required a preliminary knowledge of only the ele. Oct 20, 2015 on the meromorphic continuation of the multiple lerch zeta functions and an idea of ramanujan article pdf available october 2015 with 52 reads how we measure reads. An elementary approach to the meromorphic continuation of some. Meromorphic continuation of zeta functions associated to elliptic operators nigel higson. Here local means relative to a prime p in z, or in some ring of integers in a number eld. On analytic continuation of various multiple zetafunctions. In this case, one expects the zeta function to be a rational function, in a suitable variable. The prime number theorem, hardys theorem on the zeros of. Below, we will present all the fundamental properties of this function, and prove. Introduction it is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of.
The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of the ruelle zeta function for c1anosov ows. This writeup gives riemanns argument that the closely related function. Analytic continuation of multiple hurwitz zeta functions mehta, jay and viswanadham, g. The hurwitz zeta function can also be written as a series involving the gamma function, as the sum of dirichlet lfunctions 8, and for positive integer inputs, in terms of the polygamma function 1.
November 19, 2006 abstract we give a new proof of the meromorphic continuation property of zeta functions associated to elliptic operators. 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. Pdf meromorphic continuation of selberg zeta functions with. Meromorphic continuation of the spectral zeta kernel. Meromorphic continuation and functional equation of gl 2 eisenstein series june 6, 2011 thus, in the region of convergence. This means that all their zeros are real or occur in conjugate pairs. How does one motivate the analytic continuation of the.
Riemann showed that the function s extends from that halfplane to a meromorphic function on all of c the \riemann zeta. The riemann zeta function is defined for res 1 as follows. These are analytic objects complex functions that are intimately related to the global elds we have been studying. Analytic continuation of multiple hurwitz zeta functions. Introduction it is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of the casimir effect and boseeinstein condensation. Dynamical zeta functions for anosov flows via microlocal analysis semyon dyatlov and maciej zworski abstract. The analytic continuation and the asymptotic behaviour of certain multiple zetafunctions iii kohji matsumoto dedicated to professor akio fujii, my. What are zeta functions of graphs and what are they good for. 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. We want do derive this result using tools like the theta function. On the meromorphic continuation of the multiple lerch zeta. The field of study where the term is used and the specific meaning of the term changed in the 20th century. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.
Visualizing the riemann hypothesis and analytic continuation. A very natural question which then presents itself is the following. Again, the model is provided by the riemann zeta function. Accurately approximating the hurwitz zeta function is thus signi cant due to its important applications. Meromorphic continuation and functional equation the facts that zis essentially the mellin transform of and that is a modular form quickly give rise to the meromorphic continuation and functional equation. In this paper we prove in particular the meromorphic continuation to the entire complex plane of the zeta function of a product of a hilbert modular s. A window into zeta and modular physics msri publications volume 57, 2010 basic zeta functions and some applications in physics klaus kirsten 1. Meromorphic continuation of zeta functions associated to. Notes on the riemann zeta function january 25, 2007 1 the zeta function 1. Wewillsaymoreabout thesizeofrx for irregular graphs later in this section. The meromorphic continuation of the zeta function of siegel modular threefolds over totally real fields virdol, cristian, functiones et approximatio commentarii mathematici, 2012.
On analytic continuation of various multiple zeta functions kohji matsumoto abstract in this article we describe the development of the problem of analytic continuation of multiple zeta functions. Zeta functions in algebraic geometry mircea mustat. The meromorphic extension of the zeta function to the complex plane. Mellin, and then discuss the euler sum and its multivariable generalization. Also, for any function h2l1r and any positive number r, the fourier transform of the dilated function h rx hxr is ch rx r 1h. The argument adapts easily to the hypoelliptic operators considered by connes and moscovici in their recent work on transverse index theory in noncommutative geometry. Pdf analytic continuation of some zeta functions researchgate. The riemann zetafunction s is a meromorphic function. The meromorphic continuation of the zeta function of a. What is the analytic continuation of the riemann zeta function. Analytic continuation of multiple hurwitz zeta functions 3 the meromorphic continuation of.
The idea applies to any zeta or lfunction for which we know an analytic continuation and other reasonable properties. Riemanns second proof of the analytic continuation of the. Zeta functions for the riemann zeros by andre voros 1. Riemann showed that the function s extends from that halfplane to a meromorphic function on all of c the \riemann zeta function, analytic except for a simple pole at s 1. 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. 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.
Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent. It took 40 years for hadamard 1893, vonmangoldt 1895, and others to complete riemanns 18578 sketch of the explicit formula relating primes to zeros of the eulerriemann zeta function. We now state and prove our result on the riemann zeta function. The zeta function is an important function in mathematics. Pdf on the meromorphic continuation of the multiple lerch. Pdf meromorphic continuation of selberg zeta functions.
These are analytic objects complex functions that are intimately related to the global elds. It is one of the most important and ubiquitous special functions in mathematics, with applications in combinatorics, probability, number theory, di erential equations, etc. Analytic continuation of the riemann zeta function 9. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent the stepwise continuation technique may, however. June 6, 2011 meromorphic continuation and functional. 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 zeta function and its relation to the prime number theorem ben rifferreinert abstract.
Pdf this is an expository paper on the meromorphic continuation of zeta functions with euler products for example zeta functions of groups. Introduction to analytic number theory the riemann zeta. More general results have been recently proved by giuliettiliveranipollicott gilipo but. Selberg zeta function, meromorphic continuation, nonunitary representation, nonexpanding cusp monodromy, thermodynamic formalism, transfer operator. On analytic continuation of various multiple zetafunctions kohji matsumoto abstract in this article we describe the development of the problem of analytic continuation of multiple zetafunctions. The meromorphic continuation of a zeta function of weil and igusa type. Abstract we give a new proof of the meromorphic continuation property of zeta functions associated to elliptic operators. Meromorphic continuation of selberg zeta functions with twists having nonexpanding cusp monodromy ksenia fedosova and anke pohl abstract. We investigate meromorphic continuation of local zeta functions and properties of their poles. The main motivation was that the meromorphic continuation of archimedean local zeta functions implies the existence of fundamental solutions i. The image of this function was called an automorphism of g. June 6, 2011 meromorphic continuation and functional equation of gl. Basic zeta functions and some applications in physics. The meromorphic continuation of the zeta function of.
On the meromorphic continuation of the multiple lerch zeta functions and an idea of ramanujan article pdf available october 2015 with 52 reads how we measure reads. In the 1930s, in group theory, a meromorphic function or meromorph was a function from a group g into itself that preserved the product on the group. The meromorphic continuation of the local zeta functions was established, independently, by atiyah 4 and bernstein 6 see also 31, theorem 5. The fact that this function is analytic in this region of the complex plane is a consequence of the following basic fact.
Meromorphic continuation of dynamical zeta functions via. Riemanns second proof of the analytic continuation of the riemann zeta function andreas steiger seminar on modular forms, winter term 2006 1 abstract the riemann zetafunction. And you know the drill with youtube, if you want to stay posted about new videos, subscribe, and click the bell to. The meromorphic continuation of a zeta function of weil and. The meromorphic continuation of the zeta function of siegel modular threefolds over totally real fields.
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