Is this graph based on rationals familiar?

Has anyone come across a graph like this?

The black circles represent rationals in $(0,1)$ and their heights are roughly proportional to the reciprocal of the square of their lowest terms denominator. The red lines are drawn by eye on the pattern of the black dots.

This came from trying to create a probability distribution on the rationals where $$\Pr\left(X = \frac{a}{b}\right) = \frac{\zeta(k)}{\zeta(k-1) - \zeta(k) } \left(\frac{1}{b}\right)^k$$ where $0 \lt a \lt b$ with $a$ and $b$ coprime and where $k \gt 2$.

The red lines look somewhat like the left half of the Stern-Brocot Tree except that points with different denominators are at different heights.

Solution 1:

As Gerry mentions, this is (roughly) the square of Thomae's function, also known as the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function, the Riemann function, or the Stars over Babylon...

Solution 2:

It is surprising that no one has mentioned Ford circles. The heights of the centers of Ford circles are exactly proportional to the reciprocals of the denominators.

Let us suppose that the content of the section on total area of Ford circles is correct. Then if a point is randomly chosen in the interior of one of the Ford circles corresponding to a rational number in the interval $[0,1]$, the probability of its being in any region proportional to the region's area, then the probability that it is in the circle corresponding to a particular rational number with denominator $k$ is $$ \frac{\zeta(4)}{\zeta(3)} \left(\frac 1 k\right)^4. $$