Ytukyg

This question is related to the $g_i(x)$ functions which are defined below both in expanded form and in terms of the $f_i(x)$ functions defined in my previous question at ref(1) where all $f_i(x)$ functions are evaluated with the coefficient function $a(n)=1$. The definitions below are for $x>0$, and assume the definition $g_i(x)=0$ for $xle 0$. Note $g_2(x)=g_1left(frac{x}{sqrt{pi}}right)$ and $g_6(x)=g_5left(frac{x}{sqrt{pi}}right)$.

(1) $quad g_1(x)=frac{2}{sqrt{pi}},f_1'(x)=frac{4}{sqrt{pi},x^3}sum_limits{n=1}^infty n^2,e^{-frac{n^2}{x^2}},,qquadqquadqquadquad x>0$

(2) $quad g_1'(x)=frac{2}{sqrt{pi}},f_1''(x)=frac{4}{sqrt{pi},x^6}sumlimits_{n=1}^infty n^2 left(2,n^2-3,x^2right) e^{-frac{n^2}{x^2}},,quad x>0$

(3) $quad g_2(x)=2,f_2'(x)=frac{4,pi}{x^3}sumlimits_{n=1}^infty n^2,e^{-frac{pi,n^2}{x^2}},,qquadqquadqquadqquad x>0$

(4) $quad g_2'(x)=2,f_2''(x)=frac{4,pi}{x^6}sumlimits_{n=1}^infty n^2left(2,pi,n^2-3,x^2right) e^{-frac{pi,n^2}{x^2}},,qquad x>0$

(5) $quad g_5(x)=2,f_5(x)=2sumlimits_{n=1}^inftyleft(frac{2,n^2}{x^2}-1right),e^{-frac{n^2}{x^2}},,qquadqquadqquad x>0$

(6) $quad g_5'(x)=2,f_5'(x)=frac{4}{x^5}sumlimits_{n=1}^infty n^2,left(2,n^2-3,x^2right),e^{-frac{n^2}{x^2}},,qquadquad x>0$

(7) $quad g_6(x)=2,f_6(x)=2sumlimits_{n=1}^inftyleft(frac{2,pi,n^2}{x^2}-1right),e^{-frac{pi,n^2}{x^2}},,qquadqquadquad x>0$

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

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

(9) $quadfrac{sqrt{pi}}{2},sintlimits_0^infty g_1(x),x^{-s-1},dx=pi^{frac{s+1}{2}},xi(s+1),,quadRe(s)>0$

(10) $quadfrac{1}{2},sintlimits_0^infty g_2(x),x^{-s-1},dx=xi(s+1),,qquadqquadRe(s)>0$

(11) $quadfrac{1}{2},sintlimits_0^infty g_5(x),x^{-s-1},dx=pi^{frac{s}{2}},xi(s),,qquadqquad,,Re(s)>1$

(12) $quadfrac{1}{2},sintlimits_0^infty g_6(x),x^{-s-1},dx=xi(s),,qquadqquadqquadRe(s)>1$

The following plot illustrates the $g_1(x)$, $g_2(x)$, $g_5(x)$, and $g_6(x)$ functions defined above in blue, orange, green, and red respectively where the series for each function is evaluated over the first $1,000$ terms.

Figure (1): Illustration of $g_1(x)$, $g_2(x)$, $g_5(x)$, and $g_6(x)$ (blue, orange, green, and red)

The following plot illustrates the $g_1'(x)$, $g_2'(x)$, $g_5'(x)$, and $g_6'(x)$ functions defined above in blue, orange, green, and red respectively where the series for each function is again evaluated over the first $1,000$ terms.

Figure (2): Illustration of $g_1'(x)$, $g_2'(x)$, $g_5'(x)$, and $g_6'(x)$ (blue, orange, green, and red)

Note all four of the $g_i(x)$ functions illustrated in Figure (1) above seem to have the properties of a Cumulative Distribution Function (CDF), and all four of the corresponding $g_i'(x)$ functions illustrated in Figure (2) above seem have the properties of the corresponding Probability Density Function (PDF).

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

Question (2): When each $g_i(x)$/$g_i'(x)$ function pair is interpreted as a CDF/PDF function pair, what probability does each of these function pairs represent?

ref(1): Questions related to $f(x)$ where the Riemann Xi function $xi(s)=sintlimits_0^infty f(x),x^{-s-1},dx$