Reimann Zeta Distribution Model

Suppose an event can occur several times within a given unit of time. When the total number of occurrences of the event is unknown, we can think of it as a random variable. When a random variable X takes on values on discrete time interval from 1 to infinity, one choice of a probability density is the Reimann Zeta distribution whose probability density function is given by as follows.

[Tex]f(x) = \frac{1}{\zeta(\alpha+1)}x^{-(\alpha+1)}[/Tex]

Above expression will be applicable only when given below condition will follow.

x = 1,2,3,....[Tex]    [/Tex].
f(x) = 0, Otherwise

Where, [Tex]\alpha    [/Tex] is the parameter and [Tex]\zeta(\alpha+1)    [/Tex] is the value of the zeta function, defined by as follows.

[Tex]\zeta(y) = 1+(\frac{1}{2})^y+(\frac{1}{3})^y+(\frac{1}{4})^y + ...... + (\frac{1}{n})^y = \Sigma_{k=1}^{\infty} k^{-y}[/Tex]

The random variable X following Reimann Zeta Distribution is represented as follows.

X ~ RIE([Tex]\alpha    [/Tex])

Expected Value :

The Expected Value of the Reimann Zeta distribution can be found by summing up products of Values with their respective probabilities as follows.

[Tex]\mu = E(X) = \Sigma_{x=-\infty}^{\infty} x.f(x)[/Tex]
[Tex]\mu = \Sigma_{x=1}^{\infty} x.\frac{1}{\zeta(\alpha+1)}.x^{-(\alpha+1)}[/Tex]
[Tex]\mu = \frac{1}{\zeta(\alpha+1)}\Sigma_{x=1}^{\infty} x^{-\alpha}[/Tex]

Using the property [Tex]\zeta(y) = \Sigma_{k=1}^{\infty} k^{-y}    [/Tex], we get the following expression as follows.

[Tex]\mu = \frac{\zeta(\alpha)}{\zeta(\alpha+1)}[/Tex]

Variance and Standard Deviation : 

The Variance of the Riemann Zeta distribution can be found using the Variance Formula as follows.

[Tex]σ^2 = E( X − μ )^2 = E( X^2 ) − μ^2[/Tex]
[Tex]E(X^2) = \Sigma^{\infty}_{x=-\infty} x^2.f(x) [/Tex]
[Tex]E(X^2) = \frac{1}{\zeta(\alpha+1)} \Sigma_{x=1}^{\infty} x^2.x^{-(\alpha+1)}[/Tex]
[Tex]E(X^2) = \frac{1}{\zeta(\alpha+1)} \Sigma_{x=1}^{\infty} x^{1-\alpha}[/Tex]

Using the property [Tex]\zeta(y) = \Sigma_{k=1}^{\infty} k^{-y}    [/Tex], we get the following expression as follows.

[Tex]E(X^2) = \frac{\zeta(\alpha-1)}{\zeta(\alpha+1)}[/Tex]
[Tex]So, Var(X) = E(X^2) - \mu^2[/Tex]
[Tex]Var (X) = (\frac{\zeta(\alpha-1)}{\zeta(\alpha+1)})^2 - (\frac{\zeta(\alpha)}{\zeta(\alpha+1)})^2[/Tex]
[Tex]Var(X) = \sigma^2 = \frac{[\zeta(\alpha-1)]^2 - [\zeta(\alpha)]^2}{[\zeta(\alpha+1)]^2}[/Tex]

Standard Deviation is given by as follows.

[Tex]\sigma = \frac{1}{\zeta(\alpha+1)} \sqrt{[\zeta(\alpha-1)]^2 - [\zeta(\alpha)]^2}[/Tex]

The Riemann Zeta distribution is often used in number theory. It is not commonly referred to as a “distribution” in the statistical sense but is linked to the Riemann Zeta function. It plays a crucial role in the distribution of prime numbers.