Editing Zeta distribution (section)

== Moments ==

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

:<math>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>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>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>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.