Editing Zipf's law (section)

==Formal definition==
{{Probability distribution
| name        = Zipf's law
| type        = mass
| pdf_image   = Zipf distribution PMF.png
| pdf_caption = Plot of the Zipf PMF for {{nobr|{{math|''N'' {{=}} 10}}}}. Zipf PMF for {{nobr|{{math|''N'' {{=}} 10}} }} on a log–log scale. The horizontal axis is the index {{mvar|k}}&nbsp;. (The function is only defined at integer values of {{mvar|k}}&nbsp;. The connecting lines are only visual guides; they do not indicate continuity.)
| cdf_image   = Zipf distribution CMF.png
| cdf_caption = Plot of the Zipf CDF for {{mvar|N}} = 10. Zipf CDF for  {{nobr|{{math|''N'' {{=}} 10}} .}} The horizontal axis is the index {{mvar|k}}&nbsp;. (The function is only defined at integer values of {{mvar|k}}&nbsp;. The connecting lines do not indicate continuity.)
| parameters  = {{ubl
  | <math>s \geq 0\,</math> ([[real number|real]])
  | <math>N \in \{1,2,3\ldots\}</math> ([[integer]])
  }}
| support     = <math>k \in \{1,2,\ldots,N\}</math>
| pdf         = <math>\frac{1/k^s}{H_{N,s}}</math> where ''H<sub>N,s</sub>'' is the ''N''th generalized [[harmonic number]]
| cdf         = <math>\frac{H_{k,s}}{H_{N,s}}</math>
| mean        = <math>\frac{H_{N,s-1}}{H_{N,s}}</math>
| median      = 
| mode        = <math>1\,</math>
| variance    = <math>\frac{H_{N,s-2}}{H_{N,s}}-\frac{H^2_{N,s-1}}{H^2_{N,s}}</math>
| skewness    = 
| kurtosis    = 
| entropy     = <math>\frac{s}{H_{N,s}}\sum\limits_{k=1}^N\frac{\ln(k)}{k^s}
+\ln(H_{N,s})</math>
| mgf         = <math>\frac{1}{H_{N,s}}\sum\limits_{n=1}^N \frac{e^{nt}}{n^s}</math>
| char        = <math>\frac{1}{H_{N,s}}\sum\limits_{n=1}^N \frac{e^{int}}{n^s}</math>
}}

Formally, the Zipf distribution on {{mvar|N}} elements assigns to the element of rank {{mvar|k}} (counting from 1) the probability:

<math display="block">\ f(k;N)  ~=~ 
  \begin{cases}
     \frac{ 1 }{\ H_N }\ \frac{1}{\ k\ }\ , &\ \mbox{ if }\ 1 \le k \le N ~, \\
      {} \\
      ~~ 0 ~~ , &\ \mbox{ if }\ k < 1\ \mbox{ or }\  N < k ~.
  \end{cases}
</math> where {{mvar|H}}<sub>{{mvar|N}}</sub> is a normalization constant: The {{mvar|N}}th [[harmonic number]]:

<math display="block"> H_N \equiv \sum_{k=1}^N \frac{\ 1\ }{ k } ~.</math>

The distribution is sometimes generalized to an inverse power law with exponent {{mvar|s}} instead of {{math| 1&nbsp;.}}<ref name=adam2000/>  Namely,

<math display="block">f(k;N,s) = \frac{1}{H_{N,s}}\,\frac{1}{k^s}</math>

where {{mvar|H}}<sub>{{mvar|N}},{{mvar|s}}</sub> is a [[generalized harmonic number]]

<math display="block"> H_{N,s} = \sum_{k=1}^N \frac{1}{k^s} ~.</math>

The generalized Zipf distribution can be extended to infinitely many items ({{mvar|N}} = ∞) only if the exponent {{mvar|s}} exceeds {{math| 1&nbsp;.}} In that case, the normalization constant {{mvar|H}}<sub>{{mvar|N}},{{mvar|s}}</sub> becomes [[Riemann zeta function|Riemann's zeta function]],

<math display="block">\zeta (s) = \sum_{k=1}^\infty \frac{1}{k^s} < \infty ~.</math>

The infinite item case is characterized by the [[Zeta distribution]] and is called [[Lotka's law]]. If the exponent {{mvar|s}} is {{math| 1 }} or less, the normalization constant {{mvar|H}}<sub>{{mvar|N}},{{mvar|s}}</sub> diverges as {{mvar|N}} tends to infinity.