Possible way to plot the solution density of diophantine equations

Well, I’m trying to investigate the density of solutions to Diophantine equations. What is a general method to describe that density?

I’ve two functions:

$$varphileft(text{a},text{b},text{c}right)=thetaleft(text{a},text{b},text{c}right)tag1$$

I will look for solutions in the following range: $text{a}=left{text{a}_0,dots,text{a}_text{n}right}$ and $y=left{text{b}_0,dots,text{b}_text{m}right}$. So I will choose $text{a}_0$ as starting value and let $text{b}$ run from $text{b}_0$ until $text{b}_text{m}$ and check if there follows a $text{c}inmathbb{Z}$. Now, if I let $text{b}_0$ and $text{b}_text{m}$ constant what should I plot on the x-axis if I want to plot the solution density on the y-axis?

Example and my work:

I’ve the Diophantine equation:

$$x^2+left(frac{y}{3}right)^2=ztag2$$

In order to solve it $x$, $y$ and $z$ has to be integers.

I will look for solutions in the following range: $x=left{5,dots,7right}$ and $y=left{-10,dots,10right}$. Now I found $text{p}=21$ soltuions.

I used Mathematica to check them:

So, I should say that the density is given by:

$$rho=frac{text{# solutions}}{text{total possible solutions}}=frac{21}{left(1+7-5right)left(1+10-left(-10right)right)}=frac{21}{63}=frac{1}{3}tag3$$

Question: the general way of finding the solution density can be written as:

$$rho=frac{text{# solutions}}{left(1+x_text{n}-x_0right)left(1+y_text{m}-y_0right)}tag4$$

If I want to plot the function of $rho$ on the y-axis what should be a smart choice to put on the x-axis? Assuming that I let $y_0$ and $y_text{m}$ constant.