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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.


Illustration of g_6(x)

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.


Illustration of g_6'(x)

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?


ref(1): Questions related to $f(x)$ where the Riemann Xi function $\xi(s)=s\int\limits_0^\infty f(x)\,x^{-s-1}\,dx$


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

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