View source for Zeta distribution

{{Short description|Probability distribution in mathematics}}
{{More citations needed|date=August 2011}}
{{Probability distribution|
  name       =zeta|
  type       =mass|
  pdf_image  =[[Image:Zeta distribution PMF.png|325px|Plot of the Zeta PMF]]<br /><small>Plot of the Zeta PMF on a log-log scale. (The function is only defined at positive integer values of ''k''. The connecting lines do not indicate continuity.)</small>|
  cdf_image  =[[Image:Zeta distribution CMF.png|325px|Plot of the Zeta CMF]]|
  parameters =<math>s\in(1,\infty)</math>|
  support    =<math>k \in \{1,2,\ldots\}</math>|
  pdf        =<math>\frac{1/k^s}{\zeta(s)}</math>|
  cdf        =<math>\frac{H_{k,s}}{\zeta(s)}</math>|
  mean       =<math>\frac{\zeta(s-1)}{\zeta(s)}~\textrm{for}~s>2</math>|
  median     =|
  mode       =<math>1\,</math>|
  variance   =<math>\frac{\zeta(s)\zeta(s-2) - \zeta(s-1)^2}{\zeta(s)^2}~\textrm{for}~s>3</math>|
  skewness   =|
  kurtosis   =|
  entropy    =<math>\sum_{k=1}^\infty\frac{\log (k^s \zeta(s))}{k^s\zeta(s)}.</math>|
  mgf        =does not exist|
  char       =<math>\frac{\operatorname{Li}_s(e^{it})}{\zeta(s)}</math>|
  pgf        =<math>\frac{\operatorname{Li}_s(z)}{\zeta(s)}</math> 
}}

In [[probability theory]] and [[statistics]], the '''zeta distribution''' is a discrete [[probability distribution]]. If ''X'' is a zeta-distributed [[random variable]] with parameter ''s'', then the probability that ''X'' takes the positive integer value ''k'' is given by the [[probability mass function]]

<math display="block">f_s(k) = \frac{k^{-s}}{\zeta(s)} </math>

where ''ζ''(''s'') is the [[Riemann zeta function]] (which is undefined for ''s'' = 1).

The multiplicities of distinct [[prime factor]]s of ''X'' are [[statistical independence|independent]] [[random variable]]s.

The  [[Riemann zeta function]] being the sum of all terms <math>k^{-s}</math> for positive integer ''k'', it appears thus as the normalization of the [[Zipf's law|Zipf distribution]]. The terms "Zipf distribution" and "zeta distribution" are often used interchangeably. But while the Zeta distribution is a [[probability distribution]] by itself, it is not associated with [[Zipf's law]] with the same exponent. 

== Definition ==
The Zeta distribution is defined for positive integers <math>k \geq 1</math>, and its probability mass function is given by
<math display="block"> P(x=k) = \frac 1 {\zeta(s)} k^{-s}, </math>
where <math>s>1</math> is the parameter, and <math>\zeta(s)</math> is the [[Riemann zeta function]].

The cumulative distribution function is given by
<math display="block">P(x \leq k) = \frac{H_{k,s}}{\zeta(s)},</math>
where <math>H_{k,s}</math> is the generalized [[harmonic number]]
<math display="block">H_{k,s} = \sum_{i=1}^k \frac 1 {i^s}.</math>

== Moments ==

The ''n''th raw [[moment (mathematics)|moment]] is defined as the expected value of ''X''<sup>''n''</sup>:

<math display="block">m_n = E(X^n) = \frac{1}{\zeta(s)}\sum_{k=1}^\infty \frac{1}{k^{s-n}}</math>

The series on the right is just a series representation of the Riemann zeta function, but it only converges for values of <math>s-n</math> that are greater than unity. Thus:

<math display="block">m_n =
\begin{cases}
\zeta(s-n)/\zeta(s) & \text{for } n < s-1 \\
\infty & \text{for } n \ge s-1
\end{cases}
</math>

The ratio of the zeta functions is well-defined, even for ''n''&nbsp;&gt;&nbsp;''s''&nbsp;&minus;&nbsp;1 because the series representation of the zeta function can be [[analytic continuation|analytically continued]]. This does not change the fact that the moments are specified by the series itself, and are therefore undefined for large ''n''.

=== Moment generating function ===

The [[moment generating function]] is defined as

<math display="block">M(t;s) = E(e^{tX}) = \frac{1}{\zeta(s)} \sum_{k=1}^\infty \frac{e^{tk}}{k^s}.</math>

The series is just the definition of the [[polylogarithm]], valid for <math>e^t<1</math> so that

<math display="block">M(t;s) = \frac{\operatorname{Li}_s(e^t)}{\zeta(s)}\text{ for }t<0.</math> 

Since this does not converge on an open interval containing <math> t=0</math>, the moment generating function does not exist.

==The case ''s'' = 1==

''ζ''(1) is infinite as the [[harmonic series (mathematics)|harmonic series]], and so the case when ''s'' = 1 is not meaningful.  However, if ''A'' is any set of positive integers that has a density, i.e. if

<math display="block">\lim_{n\to\infty}\frac{N(A,n)}{n}</math>

exists where ''N''(''A'',&nbsp;''n'') is the number of members of ''A'' less than or equal to ''n'', then

<math display="block">\lim_{s\to 1^+}P(X\in A)\,</math>

is equal to that density.

The latter limit can also exist in some cases in which ''A'' does not have a density.  For example, if ''A'' is the set of all positive integers whose first digit is ''d'', then ''A'' has no density, but nonetheless, the second limit given above exists and is proportional to

<math display="block">\log(d+1) - \log(d) = \log\left(1+\frac{1}{d}\right),\,</math>

which is [[Benford's law]].

==Infinite divisibility==

The Zeta distribution can be constructed with a sequence of independent random variables with a [[geometric distribution]]. Let <math>p</math> be a [[prime number]] and <math>X(p^{-s})</math> be a random variable with a geometric distribution of parameter <math>p^{-s}</math>, namely

<math display="block"> \mathbb{P}\left( X(p^{-s}) = k \right) = p^{-ks } (1 - p^{-s} )</math>

If the random variables <math>( X(p^{-s}) )_{p \in \mathcal{P} }</math> are independent, then, the random variable <math>Z_s</math> defined by

<math display="block"> Z_s = \prod_{p \in \mathcal{P} } p^{ X(p^{-s}) }</math>

has the zeta distribution: <math>\mathbb{P}\left( Z_s = n \right) = \frac{1}{ n^s \zeta(s) }</math>.

Stated differently, the random variable <math>\log(Z_s) = \sum_{p \in \mathcal{P} }  X(p^{-s}) \, \log(p)</math> is [[Infinitely divisible distribution|infinitely divisible]] with [[Lévy measure]] given by the following sum of [[Dirac measure|Dirac masses]]:

<math display="block"> \Pi_s(dx) = \sum_{p \in \mathcal{P} } \sum_{k \geqslant 1 } \frac{p^{-k s}}{k} \delta_{k \log(p) }(dx)</math>

== See also ==

Other "power-law" distributions

*[[Cauchy distribution]]
*[[Lévy distribution]]
*[[Lévy skew alpha-stable distribution]]
*[[Pareto distribution]]
*[[Zipf's law]]
*[[Zipf–Mandelbrot law]]
*[[Infinitely divisible distribution]]
*[[Yule–Simon distribution]]

== External links ==

* {{cite CiteSeerX | title = Some remarks on the Riemann zeta distribution | citeseerx = 10.1.1.66.3284 | first = Allan | last = Gut }}  What Gut calls the "Riemann zeta distribution" is actually the probability distribution of &minus;log&nbsp;''X'', where ''X'' is a random variable with what this article calls the zeta distribution.
* {{MathWorld |title=Zipf Distribution |id=ZipfDistribution}}

{{ProbDistributions|Zeta distribution}}

{{DEFAULTSORT:Zeta Distribution}}
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