# Question on Probability Distributions Related to the Riemann Xi Function $\xi(s)$

This question is related to the $$g_6(x)$$ and $$g_6'(x)$$ functions defined below first in terms of the $$f_6(x)$$ and $$f_6'(x)$$ functions defined in my previous question at ref(1) where the $$f_6(x)$$ and $$f_6'(x)$$ functions are evaluated with the coefficient function $$a(n)=1$$ and second in expanded form. The definitions below are for $$x>0$$, and assume the definitions $$g_6(x)=0$$ and $$g_6'(x)=0$$ for $$x\le 0$$.

(1) $$\quad g_6(x)=2\,f_6(x)=2\sum\limits_{n=1}^\infty\left(\frac{2\,\pi\,n^2}{x^2}-1\right)\,e^{-\frac{\pi\,n^2}{x^2}}\,,\qquad\qquad\quad x>0$$

(2) $$\quad g_6'(x)=2\,f_6'(x)=\frac{4\,\pi}{x^5}\sum\limits_{n=1}^\infty n^2\,\left(2\,\pi\,n^2-3\,x^2\right)\,e^{-\frac{\pi\,n^2}{x^2}}\,,\qquad x>0$$

The $$g_6(x)$$ and $$g_6'(x)$$ functions defined above are related to the Riemann Xi function $$\xi(s)$$ as follows.

(3) $$\quad\frac{1}{2}\,s\int\limits_0^\infty g_6(x)\,x^{-s-1}\,dx=\xi(s)\,,\quad\Re(s)>1$$

(4) $$\quad\frac{1}{2}\,\int\limits_0^\infty g_6'(x)\,x^{-s}\,dx=\xi(s)\,,\qquad\Re(s)>1$$

The following plot illustrates the $$g_6(x)$$ function defined in (1) above where the series for $$g_6(x)$$ is evaluated over the first $$1,000$$ terms.

Figure (1): Illustration of $$g_6(x)$$

The following plot illustrates the $$g_6'(x)$$ function defined in (2) above where the series for the $$g_6'(x)$$ function is again evaluated over the first $$1,000$$ terms.

Figure (2): Illustration of $$g_6'(x)$$

Note the $$g_6(x)$$ function illustrated in Figure (1) above seems to have the properties of a Cumulative Distribution Function (CDF), and the corresponding $$g_6'(x)$$ function illustrated in Figure (2) above seems have the properties of the corresponding Probability Density Function (PDF).

Question (1): Can the $$g_6(x)$$/$$g_6'(x)$$ function pair illustrated in Figures (1) and (2) above be defined in terms of some known type of probability distribution. For example, can this function pair be represented by a generalized Gamma distribution for some $$\alpha$$, $$\beta$$, $$\gamma$$, and $$\mu$$ (see Wolfram Language GammaDistribution)?

Question (2): When the $$g_6(x)$$/$$g_6'(x)$$ function pair is interpreted as a CDF/PDF function pair, what probability does this function pair represent?

• Can you delete all the unnecessary content and leave only one function and make the main question clear – reuns Nov 30 '18 at 18:39