Editing Zipf's law (section)

==Related laws==
A generalization of Zipf's law is the [[Zipf–Mandelbrot law]], proposed by [[Benoit Mandelbrot]], whose frequencies are:

<math display="block">f(k;N,q,s) = \frac{1}{\ C\ }\ \frac{ 1 }{\ \left( k + q\right)^s} ~.</math>{{Clarify|date=September 2023|reason=denominator missing from equation}}

The constant {{mvar|C}} is the [[Hurwitz zeta function]] evaluated at {{mvar|s}}.

Zipfian distributions can be obtained from [[Pareto distribution]]s by an exchange of variables.<ref name=adam2000/>

The Zipf distribution is sometimes called the '''discrete Pareto distribution'''<ref name=john1992/> because it is analogous to the continuous [[Pareto distribution]] in the same way that the [[Uniform distribution (discrete)|discrete uniform distribution]] is analogous to the [[Uniform distribution (continuous)|continuous uniform distribution]].

The tail frequencies of the [[Yule–Simon distribution]] are approximately

<math display="block">f(k;\rho) \approx \frac{\ [\mathsf{constant}]\ }{ k^{(\rho + 1)} }</math> for any choice of {{nobr|{{math| ''ρ'' > 0}} .}}

In the [[parabolic fractal distribution]], the logarithm of the frequency is a quadratic polynomial of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship.<ref name=Galien/> Like fractal dimension, it is possible to calculate Zipf dimension, which is a useful parameter in the analysis of texts.<ref name=efte2006/>

It has been argued that [[Benford's law]] is a special bounded case of Zipf's law,<ref name="Galien"/> with the connection between these two laws being explained by their both originating from scale invariant functional relations from statistical physics and critical phenomena.<ref name=piet2001/> The ratios of probabilities in Benford's law are not constant. The leading digits of data satisfying Zipf's law with {{nobr| {{math|s {{=}} 1}} ,}} satisfy Benford's law.
{| class="wikitable" style="text-align: center;"
|-
!<math>n</math>
!Benford's law: <math>P(n) = </math><br/><math>\log_{10}(n+1)-\log_{10}(n)</math>
!<math>\frac{\log(P(n)/P(n-1))}{\log(n/(n-1))}</math>
|-
| 1
| 0.30103000
|  
|-
| 2
| 0.17609126
| −0.7735840
|-
| 3
| 0.12493874
| −0.8463832
|-
| 4
| 0.09691001
| −0.8830605
|-
| 5
| 0.07918125
| −0.9054412
|-
| 6
| 0.06694679
| −0.9205788
|-
| 7
| 0.05799195
| −0.9315169
|-
| 8
| 0.05115252
| −0.9397966
|-
| 9
| 0.04575749
| −0.9462848
|}