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DEFINITION


The Riemann zeta-function ζ(''s'') is defined for any Complex Number ''s'' with Real Part > 1 by the Dirichlet Series :
:
\zeta(s) =
\sum_{n=1}^\infin rac{1}{n^s}

In the region {''s'' in (''s'') > 1},
this Infinite Series converges and defines a function Analytic in this region. Bernhard Riemann Realized that the zeta-function can be extended by Analytic Continuation in a unique way to a Meromorphic Function ζ(''s'') defined for all complex numbers ''s'' with ''s'' ≠ 1. It is this function that is the object of the Riemann Hypothesis .


VALUES AT THE INTEGERS

See Also: Zeta constants



The following are values of the zeta function for some small numbers.

:\zeta(1) = 1 + rac{1}{2} + rac{1}{3} + \cdots = \infty; this is the Harmonic Series .
:\zeta(2) = 1 + rac{1}{2^2} + rac{1}{3^2} + \cdots = rac{\pi^2}{6}; the demonstration of this equality is known as the Basel Problem .
:\zeta(3) = 1 + rac{1}{2^3} + rac{1}{3^3} + \cdots \approx 1.202\dots ; this is called Apéry's Constant
:\zeta(4) = 1 + rac{1}{2^4} + rac{1}{3^4} + \cdots = rac{\pi^4}{90}
:\zeta(5) = 1 + rac{1}{2^5} + rac{1}{3^5} + \cdots \approx 1.036\dots
:\zeta(6) = 1 + rac{1}{2^6} + rac{1}{3^6} + \cdots = rac{\pi^6}{945}
:\zeta(7) = 1 + rac{1}{2^7} + rac{1}{3^7} + \cdots \approx 1.0083\dots
:\zeta(8) = 1 + rac{1}{2^8} + rac{1}{3^8} + \cdots = rac{\pi^8}{9450}
:\zeta(9) = 1 + rac{1}{2^9} + rac{1}{3^9} + \cdots \approx 1.0020\dots
:\zeta(10) = 1 + rac{1}{2^{10}} + rac{1}{3^{10}} + \cdots = rac{\pi^{10}}{93555}
:\zeta(12) = 1 + rac{1}{2^{12}} + rac{1}{3^{12}} + \cdots = rac{691\pi^{12}}{638512875}
:\zeta(14) = 1 + rac{1}{2^{14}} + rac{1}{3^{14}} + \cdots = rac{2\pi^{14}}{18243225}


RELATIONSHIP TO PRIME NUMBERS


The connection between this function and Prime Number s was already realized by Leonhard Euler :
:
\zeta(s) = \prod_{p} rac{1}{1-p^{-s}}

an Infinite Product extending over all prime numbers ''p''. This is called an Euler Product formula and converges for Re(s) > 1. It is a consequence of two simple and fundamental results in mathematics; the formula for the Geometric Series and the Fundamental Theorem Of Arithmetic .


Proof of the Euler product formula


Each factor (for a given prime ''p'') in the product above can be expanded to a Geometric Series consisting of the reciprocal of ''p'' raised to multiples of ''s'', as follows

: rac{1}{1-p^{-s}} = 1 + rac{1}{p^s} + rac{1}{p^{2s}} + rac{1}{p^{3s}} + \cdots + rac{1}{p^{ks}} + \cdots







where ''B''2''k'' are the ). These give well-known infinite series for π . For odd integers the case is not so simple; Apéry proved that ζ(3) was an irrational number, and using related methods it can be shown an infinite number of other ζ-values at odd positive integers are irrational. For a discussion of ζ(s) at odd positive integers, see Zeta Constants ; for negative values, see Bernoulli Numbers . For the
zeta-function on the critical line, see Z-function .

The reciprocal of the zeta-function may be expressed as a Dirichlet Series over the Möbius Function μ(''n''):
:
rac{1}{\zeta(s)} = \sum_{n=1}^{\infin} rac{\mu(n)}{n^s}

for every complex number ''s'' with real part > 1. There are a number of similar relations involving various well-known Multiplicative Function s; these are given in the article on the Dirichlet Series .


The above, together with the expression for ζ(2), can be used to prove that the probability of two random integers being Coprime is 6/π2. The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of ''s'' is greater than 1/2.

For sums involving the zeta-function at integer values, see Rational Zeta Series .


THE RIEMANN ZETA FUNCTION AS A MELLIN TRANSFORM


The Mellin Transform of a function ''f''(''x'') is defined as
:\{ \mathcal{M} f \}(s) = \int_0^\infty f(x)x^s rac{dx}{x}
in the region where the integral is defined. There are various expressions for the zeta-function as a Mellin transform. If the real part of s is greater than one, we have

:\Gamma(s)\zeta(s) =\left\{ \mathcal{M} \left( rac{1}{\exp(x)-1} ight) ight\}(s)

By subtracting off the first terms of the power series expansion of 1/(exp(''x'') − 1) around zero, we can get the zeta-function in other regions. In particular, in the critical strip we have

:\Gamma(s)\zeta(s) = \left\{ \mathcal{M}\left( rac{1}{\exp(x)-1}- rac1x ight) ight\}(s)

and when the real part of s is between −1 and 0,

:\Gamma(s)\zeta(s) = \left\{\mathcal{M}\left( rac{1}{\exp(x)-1}- rac1x+ rac12 ight) ight\}(s)

We can also find expressions which relate to prime numbers and the Prime Number Theorem . If π(''x'') is the Prime Counting Function , then

:\log \zeta(s) = s \int_0^\infty rac{\pi(x)}{x(x^s-1)}dx

for values with \Re(s)>1. We can relate this to the Mellin transform of π(''x'') by
rac{\log \zeta(s)}{s} - \omega(s) = \left\{\mathcal{M} \pi(x) ight\}(-s)
where

:\omega(s) = \int_0^\infty rac{\pi(s)}{x^{s+1}(x^s-1)}dx

converges for \Re(s)> rac12.

A similar Mellin transform involves the Riemann prime counting function ''J''(''x''), which counts prime powers ''p''''n'' with a weight of 1/''n'', so that J(x) = \sum rac{\pi(x^{1/n})}{n}.
Now we have

: rac{\log \zeta(s)}{s} = \left\{\mathcal{M} J ight\}(-s)

These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's prime counting function is easier to work with, and π(''x'') can be recovered from it by Möbius Inversion .


SERIES EXPANSIONS

The Riemann zeta function is Meromorphic with a single pole of order one at
''s'' = 1. It can therefore be expanded as a Laurent Series about ''s'' = 1;
the series development then is

:\zeta(s) = rac{1}{s-1} + \gamma_0 + \gamma_1(s-1) + \gamma_2(s-1)^2 + \cdots.

The constants here are called the Stieltjes Constants and can be defined
as

:\gamma_k = rac{(-1)^k}{k!} \lim_{N ightarrow \infty} \left(\sum_{m \le N} rac{\ln^k m}{m} - rac{\ln^{k+1}N}{k+1} ight).

The constant term γ0 is the Euler-Mascheroni Constant .

Another series development valid for the entire complex plane is

:\zeta(s) = rac{1}{s-1} - \sum_{n=1}^\infty (\zeta(s+n)-1) rac{x^{\overline{n}}}{(n+1)!}

where x^{\overline{n}} is the Rising Factorial
x^{\overline{n}} = x(x+1)\cdots(x+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 .


GLOBALLY CONVERGENT SERIES

A globally convergent series for the zeta function valid for all complex-valued ''s'' except ''s''=1, was conjectured by Konrad Knopp and proven 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}

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 .


UNIVERSALITY

The critical strip of the Riemann zeta function has the remarkable property of universality. This Zeta-function Universality states that there exists some location on the critical strip that approximates any Holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable.


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 + \cdots, but we can re-write it as a sum of reciprocals:



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. The simplest of these are the Hurwitz Zeta Function
:\zeta(s,q) = \sum_{k=0}^\infty (k+q)^{-s},
which coincides with Riemann's zeta-function when ''q'' = 1.

The Polylogarithm is given by
:Li_s(z) \equiv \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.

The Clausen function Cl_{s} ( heta ) that can be chosen as the Real or Imaginary part of Li_{s} (e^{i heta})


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.


SEE ALSO




REFERENCES