This question is related to my Math Stack Exchange question at the following link:

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

The definitions below are for $x>0$, and assume the definition $g(x)=0$ for $x\le 0$.

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

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

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

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

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

The following two figures illustrates the $g(x)$ and $g'(x)$ functions defined above where formulas (1) and (2) are both evaluated over the first $1,000$ terms.

Illustration of g(x)

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

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

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

I've been wondering if the $g(x)$/$g'(x)$ function pair illustrated in Figures (1) and (2) above can be represented in terms of the CDF/PDF function pair corresponding to a known probability distribution type. I've noticed Mathematica supports a variety of probability distribution types and some have parameters which can be used to shape the distribution such as the generalized Gamma distribution with parameters $\alpha$, $\beta$, $\gamma$, and $\mu$ (see Wolfram Language GammaDistribution). I've also noticed Mathematica has functions for testing distribution data such as FindDistribution, FindDistributionParameters, EstimatedDistribution, and DistributionFitTest but I'm not particularly familiar with these functions.

Question: Can Mathematica be used to determine if the $g(x)/g'(x)$ function pair defined in (1) and (2) above can be represented in terms of the CDF/PDF function pair corresponding to a known probability distribution type and if so, how can this be achieved?

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    $\begingroup$ What have you tried so far? It would be helpful to add your g(x) function in Wolfram Language syntax. Some users, myself included, don't have the maths background to easily decipher and convert the formula as written. $\endgroup$ – Carl Lange May 8 '19 at 8:18
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    $\begingroup$ To save future answerers some time: g[x_]:=2 NSum[((2\[Pi] n^2)/x^2-1)E^((-\[Pi] n^2)/x^2),{n,\[Infinity]}], gp[x_]:=(4\[Pi])/x^5 NSum[n^2 (2\[Pi] n^2-3x^2)E^((-\[Pi] n^2)/x^2),{n,\[Infinity]}] $\endgroup$ – Carl Lange May 8 '19 at 9:09
  • $\begingroup$ @CarlLange Thanks for converting my formulas to Wolfram Language syntax. I plotted a few of the distribution types supported by Mathematica for manual comparison, but it seems like Mathematica has the ability to automatically fit a distribution based on data. I have formulas versus data. It appears RandomVariate can be used to generate data based on predefined distribution types supported by Mathematica, but I'm not sure how to generate data based on my formulas for $g(x)$ and $g'(x)$. $\endgroup$ – Steven Clark May 8 '19 at 14:19
  • $\begingroup$ To the physicist in me, this kinda looks like a Planck distribution. The function m/(h^(1-(1 + n)/m)Gamma[(1+n)/m]PolyLog[(1+n)/m, 1])(h x^n)/(Exp[h x^m] - 1) with n=40 and m=1/2 (with h around 80 or 81) seems to work to some extent. It is a little bit artificial though; I'm sure there are better approximations. $\endgroup$ – AccidentalFourierTransform May 8 '19 at 15:56
  • $\begingroup$ You've mentioned using "data" and "testing distribution data" but I don't think you want to go down that road. You have the exact CDF and PDF. Any approximations with other distributions (if you don't find an equivalent already-named distribution) ought to be done with the approximating CDF's and PDF's. $\endgroup$ – JimB May 8 '19 at 16:12

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