The zeta function is an important function in mathematics. In particular, it doesnt converge at any of the zeros of the zetafunction, trivial or otherwise. Grossencharacter lfunctions lfunctions for gaussian integers we try to imitate the argument used by riemann for proving the analytic continuation of s and its functional equation. Analytic continuation and the riemann zeta function ib. Moreover location of singularities of such zeta functions is. Finally in section 5, we will show that the special values of the multiple lucas zeta function at negative arguments are rational. The function gis necessarily the unique holomorphic function on that restricts to f, and we call it the analytic continuation of f to it would make more sense to call it the holomorphic continuation of fto, but as explained in remark15. To understand how the zeta function is defined for real part less than or equal to 1, you have to be familiar with analytic continuation. On analytic continuation of various multiple zetafunctions. Analytic continuation is a very important mathematical technique which allows us to extend the domain of functions. The many interesting algebraic and combinatorial aspects of the. In particular, it doesnt converge at any of the zeros of the zeta function, trivial or otherwise. By a process known as analytic continuation not discussed in the text, the function s can be extended to a function holomorphic in c 1. Analytic continuation of the riemann zeta function.
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. The riemann hypothesis is a conjecture that the riemann zeta function has its zeros only at the negative. The prime number theorem, hardys theorem on the zeros of. Nov 09, 2014 analytic continuation and the riemann zeta function. In this video, i prove an analytic continuation of the riemann zeta function for all positive rez. The riemann zeta function and its analytic continuation. This thesis is an exposition of the riemann zeta function. The extended function is also written s and for re s 1 it equals. The next proposition shows that analytic continuations are unique in. The riemann zeta function or eulerriemann zeta function. Visualizing the riemann hypothesis and analytic continuation.
Section 4 is devoted for poles and residues of the series in. Zeros of riemann zeta function university of chicago. We will also see that an similar property of ls for. A general formula for the zeros of the analytic continuation formula of the riemann zeta function is obtained and it is shown that these zeros will always be real. Following riemann paper about analytic continuation of zeta function. Analytic continuation of the multiple lucas zeta functions. In this paper, we present riemann zeta function and its analytic continuation and functional equation. The fact that riemann zeta function doesnt have a zero on res 1 is the most crucial step in the proof of the prime number theorem. Zagier in za1, and has been the focus of intense study in recent years, appearing in connection with arithmetic and hyperbolic geometry, moduli spaces, number theory, and quantum physics see e. 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 zeta function.
The function has an analytic continuation to cnf1gwith a simple pole with residue 1 at s 1. It is essential in higher level mathematics and physics and leads to some remarkable results. Analytic continuation of operators operators acting. On analytic continuation of multiple lfunctions and related. Nov 08, 2017 in this video, i prove an analytic continuation of the riemann zeta function for all positive rez. We extend riemanns own planar analytic continuation r. The riemann zeta function satisfies the reflection functional equation. Titchmarsh, the theory of the riemann zeta function. Riemanns second proof of the analytic continuation of the. As shown by riemann, s extends to c as a meromorphic. A new formula for the riemann zeta function zetas in terms of nested series of zetan is derived. From the zeros of the riemann zeta function to its analytical.
The lucasbalancing zeta function admits its analytic continuation over the whole complex plane except its poles. The zeta function and the riemann hypothesis contents. We discuss the riemann zeta function, the topology of its domain, and make an argument against the riemann hypothesis. In this paper we shall define the analytic continuation of the mul tiple eulerriemannzagier zeta functions of depth d. In this chapter, we will see a proof of the analytic continuation of the riemann zeta function. A multiple lfunction and a multiple hurwitz zeta function of eulerzagier type are introduced. Since f 1 and f 2 are analytic on u, f 1 f 2 is analytic too and we can develop it as a taylorseries around. The critical strip of the riemann zeta function has the remarkable property of universality.
We will begin with some historical background, then define of the zeta function and some important. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. The zeta function and its relation to the prime number theorem ben rifferreinert abstract. To understand how the zetafunction is defined for real part less than or equal to 1, you have to be familiar with analytic continuation. In this brief insight article the analytic continuations of the lerch transcendent and riemann zeta functions are achieved via the eulers series transformation zeta and a generalization thereof, the eprocess lerch. Pdf analytic continuation of multiple zeta functions. D with fz fz in d, then we say that fis an analytic continuation of f. The analytic continuation of the lerch and the zeta functions. We have taken as the definition of the riemann zeta function. Analytic continuation of operators operators acting complex.
Euler computed the values of the zeta function at the negative integers us ing both abel summation 75 years before abel and the euler maclaurin. Analytic continuation of the hurwitz zeta function with. If fz is analytic in a domain dand fz is analytic in a domain d0. Remember that the zeta functon is defined by the series. Pdf analytic continuation of bernoulli numbers, a new. Mellin, and then discuss the euler sum and its multivariable generalization. Zeta function regularization procedure and illustrate the result obtained by blau visser and wipf, which will be our starting point for the application of the new formulas. The fact that this function is analytic in this region of the complex plane is a consequence of the following basic fact. Chandrasekharan tata institute of fundamental research, bombay 1953. Chapter 9 analytic continuation for every complex problem, there is a solution that is simple, neat, and wrong. Find, read and cite all the research you need on researchgate. The riemann zeta function is defined as the analytic continuation of the function defined for. And you know the drill with youtube, if you want to stay posted about new videos, subscribe, and click the bell to.
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. 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. The idea applies to any zeta or lfunction for which we know an analytic continuation and other reasonable properties. Pdf analytic continuation of the multiple fibonacci zeta. We will begin with the gamma function and some properties of the gamma function. Pdf analytic continuity of the riemann zeta function researchgate. The objective of this dissertation is to study the riemann zeta function in particular it will examine its analytic continuation, functional equation and applications. 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. How does one motivate the analytic continuation of the. Riemann zeta function we will eventually deduce a functional equation, relating s to 1 s.
Analytic continuation of riemanns zeta function and values at negative integers via eulers transformation of series jonathan sondow communicated by william w. Leonhard euler considered the above series in 1740 for positive integer values of s, and later chebyshev extended the definition to re s 1. In section 3, we derive a new expression for the analytic continuation of the hurwitz zeta function, viewed as. A multiple l function and a multiple hurwitz zeta function of eulerzagier type are introduced. Zeta functions in algebraic geometry mircea mustat. In section 3, we will show the analytic continuation of the multiple lucas zeta function. A similar idea applies to any zeta or lfunction with analytic continuation, functional equation, and euler product. We prove that a series derived using eulers transformation provides the analytic continuation of s for all complex s 1. From the zeros of the riemann zeta function to its. We use an important summation formula, due to euler. Many of the results about s that can be found with the usual tools of complex analysis have very. Section 4 is devoted for poles and residues of the series in 1.
Dec 09, 2016 3blue1brown is a channel about animating math, in all senses of the word animate. Analytic continuation of riemann zeta function ijariit. What is the analytic continuation of the riemann zeta function. Dedekind zeta function os for gaussian integers o 4. 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. The riemann zetafunction s is a meromorphic function. Many people are using the term assigned value or analytic continuation for divergent series but this explantion is so lacking and can be replaced with a much easier and simpler term of explantion for me as i see it when i am looking at the zeta function i dont see or use the term assigned value or analytic continuation. While making the argument in the classical formalism, we discuss the material as it relates to the theory of in nite complexity toic. Chapter 9 the functional equation for the riemann zeta function. The riemann zeta function plays a pivotal role in analytic number theory and has applications.
Riemann showed that the function s extends from that halfplane to a meromorphic function on all of c the \riemann zeta. We know that in complex analysis 1, functions can be analytic continued from integer points non the real line to complex plane s, eg. Analytic continuation and the riemann zeta function. The aim of these lectures is to provide an intorduc tion to the theory of the riemann zetafunction for stu dents who might later want to do research on the subject. The zeta function, usually referred to as the riemann zeta.
On analytic continuation of multiple lfunctions and. Dirichlet eta function negative integer formula isaac. Again, the model is provided by the riemann zeta function. While it was not created by riemann, it is named after him because he was able to prove an important relationship between its zeros and the distribution of the prime numbers. The analytic continuation of the lerch transcendent and the riemann zeta function introduction. Analytic continuation is not some strange method to give meaning to the zeta function.
The next proposition shows that analytic continuations are unique in a certain way. Notes on the riemann zeta function january 25, 2007 1 the zeta function 1. Importance of the zeta function the zeta function is a very important function in mathematics. Then we prove that the riemann zeta function can be represented by eulers product. Included are techniques of analytic continuation and relationships to special functions. 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. Underlying it is the rather plain fact that if a holomorphic function on a connected open set in c is identically zero on some open subset then it is zero everywhere.
Moreover location of singularities of such zeta functions is studied in detail. Analytic continuation of them as complex functions of several variables is established by an application of the eulermaclaurin summation formula. The riemann zeta function is eventually obtained through this. Pdf in this paper we shall dene the analytic continuation of the mul tiple eulerriemannzagier zeta functions of depth d. The zeta function is often called the \riemann zeta function because riemann instigated serious study of it. Introduction to analytic number theory the riemann zeta. Some generalizations of the riemann zeta function are outlined, as well as the calculation of zeta constants and the development of some identities. Visualizing the riemann zeta function and analytic. There are various methods to derive this functional equation, see e. It took 40 years for hadamard 1893, vonmangoldt 1895, and. Using an irreducible using an irreducible m z,w from 1. Hence whether something can be done along these lines is immediately related to the possibility of performing analytic continuation of the logarithmic derivative of the zeta function. The first proof ot the analytic continuation of the riemann zeta. Analytic continuation of the riemann zeta function 9.
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