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In Mathematics , the Riemann zeta function, named after German mathematician Bernhard Riemann , is a Function of great significance in Number Theory because of its relation to the Distribution Of Prime Numbers . It also has applications in other areas such as Physics , Probability Theory , and applied Statistics .


DEFINITION


The Riemann zeta-function ζ(''s'') is the function of a complex variable ''s'' initially defined by the following Infinite Series :
:
\zeta(s) =
\sum_{n=1}^\infty rac{1}{n^s}

for certain values of ''s'' and then at ''s''=1. The analytic continuation process is unambiguous, resulting in a unique function, and in addition to extending ζ(''s'') beyond the domain of the convergence of the original series, Riemann established a Functional Equation for the zeta function, which relates its values at points ''s'' and 1 − ''s''. The celebrated Riemann Hypothesis , formulated in the same paper of Riemann, is concerned with zeros of this analytically extended function. To emphasize that ''s'' is viewed as a ''complex'' number, it is frequently written in the form ''s'' = σ + ''it'', where σ = Re(''s'') is the Real and ''t'' = Im(''s'') is the Imaginary part of ''s''.


EULER PRODUCT FORMULA


The connection between zeta function and Prime Number s was discovered by Leonhard Euler , who proved the identity
:
\begin{align}
\sum_{n\geq 1} rac{1}{n^s}& = \prod_{p ext{ prime}} rac{1}{1-p^{-s}} = \
& = \left(1 + rac{1}{2^s} + rac{1}{4^s} + \cdots ight) \left(1 + rac{1}{3^s} + rac{1}{9^s} + \cdots ight) \cdots \left(1 + rac{1}{p^s} + rac{1}{p^{2s}} + \cdots ight) \cdots,
\end{align}


where, by definition, the left hand side is ζ(''s'') and the Infinite Product in the right hand side extends over all prime numbers ''p'' (such expressions are called Euler Product s). Both sides of this identity converge for Re(''s'') > 1. The Proof Of Euler's Identity uses only the formula for the Geometric Series and the Fundamental Theorem Of Arithmetic . Since the Harmonic Series , obtained when ''s'' = 1, diverges, Euler's formula implies that there are infinitely many primes.

The above product can be used to show that rac 1{\zeta(n)} is the probability that ''n'' randomly selected integers are Relatively Prime .


VARIOUS PROPERTIES

For the Riemann zeta function on the critical line, see Z-function . For sums involving the zeta-function at integer values, see Rational Zeta Series .


Specific values

See Also: Zeta constant



The following are the most commonly used values of the Riemann zeta function.

:\zeta(1) = 1 + rac{1}{2} + rac{1}{3} + \cdots = \infty; this is the Harmonic Series .

:\zeta(3/2) \approx 2.612

:\zeta(2) = 1 + rac{1}{2^2} + rac{1}{3^2} + \cdots = rac{\pi^2}{6} \approx 1.645; the demonstration of this equality is known as the Basel Problem .

:\zeta(5/2) \approx 1.341

:\zeta(3) = 1 + rac{1}{2^3} + rac{1}{3^3} + \cdots \approx 1.202 ; this is called Apéry's Constant

:\zeta(7/2) \approx 1.127

:\zeta(4) = 1 + rac{1}{2^4} + rac{1}{3^4} + \cdots = rac{\pi^4}{90} \approx 1.0823


The functional equation

The zeta-function satisfies the following Functional Equation :
:
\zeta(s) = 2^s\pi^{s-1}\sin\left( rac{\pi s}{2} ight)\Gamma(1-s)\zeta(1-s)


valid for all ''s'' in \scriptstyle{C \setminus \lbrace 0,1 brace}. Here, Γ denotes the Gamma Function . This formula is used to construct the analytic continuation in the first place. At ''s'' = 1, the
zeta-function has a simple Pole with Residue 1. The equation also shows that the zeta function has trivial zeros at −2, −4, ... .

There is also a symmetric version of the functional equation, given by first defining

:\xi(s) = \pi^{-s/2}\Gamma\left( rac{s}{2} ight)\zeta(s).

The functional equation is then given by

:\xi(s) = \xi(1 - s).\

The functional equation also gives the asymptotic limit

:
\zeta \left( {1 - s} ight) = \left( { rac{s}} ight)^s \sqrt { rac{s}} \cos \left( { rac{2}} ight)\left( {1 + O\left( { rac{1}{s}} ight)} ight).

(''Gergő Nemes, 2007'')


Zeros of the Riemann zeta function

The Riemann zeta function has zeros at the negative even integers (see the functional equation). These are called the trivial zeros. They are trivial only in the sense that their existence is relatively easy to prove, for example, from the connection with the gamma function as shown below. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, opens the way to an astonishingly rich vein of mathematical inquiry. It is known that any non-trivial zero lies in the open strip {''s'' ∈ '''C''': 0 < Re(''s'') < 1}, which is called the '''critical strip'''. The Riemann Hypothesis , considered to be one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero ''s'' has Re(''s'') = 1/2. In the theory of the Riemann zeta function, the set {''s'' ∈ '''C''': Re(''s'') = 1/2} is called the '''critical line'''.






where s^{\overline{n}} is the Rising Factorial
s^{\overline{n}} = s(s+1)\cdots(s+n-1). This can be used recursively
to extend the Dirichlet series definition to all complex numbers.

The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss-Kuzmin-Wirsing Operator acting on ''x''''s''−1; that context gives rise to a series expansion in terms of the Falling Factorial .


Hadamard product


On the basis of Weierstrass's Factorization Theorem , Hadamard gave the Infinite Product expansion

:\zeta(s) = rac{e^{As}}{2(s-1)\Gamma(1+s/2)} \prod_ ho \left(1 - rac{s}{ ho} ight) e^{s/ ho}

where the product is over the non-trivial zeros ρ of ζ and

A


the letter γ again denoting the Euler-Mascheroni Constant .


Globally convergent series

A globally convergent series for the zeta function, valid for all complex numbers ''s'' except ''s'' = 1, was conjectured by Konrad Knopp and proved by Helmut Hasse in 1930 :

:\zeta(s)= rac{1}{1-2^{1-s}}
\sum_{n=0}^\infty rac {1}{2^{n+1}}
\sum_{k=0}^n (-1)^k {n \choose k} (k+1)^{-s}.

The series only appeared in an Appendix to Hasse's paper, and did not become generally known until it was rediscovered more than 60 years later (see Sondow, 1994).

Peter Borwein has shown a very rapidly convergent series suitable for high precision numerical calculations. The algorithm, making use of Chebyshev Polynomial s, is described in the article on the Dirichlet Eta Function .


APPLICATIONS


Although mathematicians regard the Riemann zeta function as being primarily relevant to the "purest" of mathematical disciplines, Number Theory , it also occurs in applied Statistics (see Zipf's Law and Zipf-Mandelbrot Law ), physics, and the mathematical theory of Musical Tuning .

During several physics-related calculations, one must evaluate the sum of the positive integers; paradoxically, on physical grounds one expects a finite answer. When this situation arises, there is typically a rigorous approach involving much in-depth analysis, as well as a "short-cut" solution relying on the Riemann zeta-function. The argument goes as follows: we wish to evaluate the sum 1 + 2 + 3 + 4 + · · · , but we can re-write it as a sum of reciprocals:

: \begin{align} S &{}=1 + 2 + 3 + 4 + \cdots \
&{}= \left( rac{1}{1} ight)^{-1} + \left( rac{1}{2} ight)^{-1} + \left( rac{1}{3} ight)^{-1} + \left( rac{1}{4} ight)^{-1} + \cdots \
&{}=\sum_{n=1}^{\infin} rac{1}{n^{-1}}. \end{align}

The sum ''S'' appears to take the form of \zeta(-1). However, −1 lies outside of the Domain for which the Dirichlet series for
the zeta-function converges. However, a Divergent Series of positive terms such as this one can sometimes be summed in a reasonable way by the method of Ramanujan Summation (see Hardy, ''Divergent Series.'') Ramanujan summation involves an application of the Euler-Maclaurin Summation Formula , and when applied to the zeta-function, it extends its definition to the whole complex plane. In particular

:1+2+3+\cdots = - rac{1}{12} (\Re)

where the notation (\Re) indicates Ramanujan summationhttp://algo.inria.fr/seminars/sem01-02/delabaere2.pdf.

For even powers we have:

:1+2^{2k}+3^{2k}+\cdots = 0 (\Re)

and for odd powers we have a relation with the Bernoulli Number s:

:1+2^{2k-1}+3^{2k-1}+\cdots = - rac{B_{2k}}{2k} (\Re).

Zeta Function Regularization is used as one possible means of Regularization of Divergent Series in Quantum Field Theory . In one notable example, the Riemann
zeta-function shows up explicitly in the calculation of the Casimir Effect .


GENERALIZATIONS

There are a number of related Zeta Function s that can be considered to be generalizations of Riemann's zeta-function. These include the Hurwitz Zeta Function

:\zeta(s,q) = \sum_{k=0}^\infty (k+q)^{-s},

which coincides with Riemann's zeta-function when ''q'' = 1 (note that the lower limit of summation in the Hurwitz zeta function is 0, not 1), the Dirichlet L-function s and the Dedekind Zeta-function . For other related functions see the articles Zeta Function and L-function .

The Polylogarithm is given by

:\mathrm{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}

which coincides with Riemann's zeta-function when ''z'' = 1.

The Lerch Transcendent is given by
:\Phi(z, s, q) = \sum_{k=0}^\infty
rac { z^k} {(k+q)^s}
which coincides with Riemann's zeta-function when ''z'' = 1 and ''q'' = 1 (note that the lower limit of summation in the Lerch transcendent is 0, not 1).

The Clausen function Cl_{s} ( heta ) that can be chosen as the real or imaginary part of \mathrm{Li}_{s} (e^{i heta})

The Multiple Zeta Functions are defined by
:\zeta(s_1,s_2,\dots,s_n) = \sum_{k_1>k_2>\dots>k_n>0} k_1^{-s_1}k_2^{-s_2}\cdots k_n^{-s_n}.
One can analytically continue these functions to the n-dimensional complex space. The special values of these functions are called Multiple Zeta Values by number theorists and have been connected to many different branches in mathematics and physics.


ZETA-FUNCTIONS IN FICTION


Neal Stephenson 's 1999 novel '' Cryptonomicon '' mentions the zeta-function as a Pseudo-random Number source, a useful component in Cipher design.

The zeta-function is a major part of the plot of Thomas Pynchon 's novel '' Against The Day '' ( 2006 ).

The popular T.V. Show NUMB3RS had an episode ("Prime Suspect") in which criminals kidnapped a child and demanded as ransom a possible proof of the Riemann Hypothesis from a mathematician. The proof would be used to steal interest rates from an encrypted website.


SEE ALSO




NOTES



REFERENCES