Discrete probability distribution involving curtailed Riemann zeta values $\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Var}{\operatorname{Var}}$We define the discrete random variable $X$ as having the probability mass function $$f_{X}(k) = \Pr(X=k) = \zeta(k)-1, $$ for $k \geq 2 $.
Here, $\zeta(\cdot)$ is the Riemann zeta function, defined as $$\zeta(s) = \sum_{n=1}^{\infty} n^{-s} $$ for $\Re(s) >1 $.
Now, $X$ is indeed a discrete RV, as we have $$\sum_{k=2}^{\infty} p_k = \sum_{k=2}^{\infty} (\zeta(k)-1) = 1,$$ (which we can find, for example, here) and it is clear that for all $k$ it holds that $$0 \leq p_k \leq 1 .$$
Furthermore, we can find the first and second moments of $X$. The mean amounts to $$E[X] = \sum_{k=2}^{\infty} k \big(\zeta(k)-1\big) = 1+\frac{\pi^{2}}{6} .$$
Moreover, we have $$E[X^{2}] = \sum_{k=2}^{\infty} k^{2} \big( \zeta(k)-1 \big) = 1 + \frac{\pi^{2}}{2} + 2 \zeta(3), $$ so we obtain \begin{align*} \Var(X) &= E[X^2] - E[X]^{2} \\ 
&= \frac{\pi^2}{6} +2 \zeta(3) - \frac{\pi^4}{36} \\ 
&=  \zeta(2) + 2 \zeta(3) - \frac{5}{2} \zeta(4). \end{align*}
Question: does this discrete random variable involving curtailed Riemann zeta values come up in the literature on probability theory and/or statistics? Does it have any applications?
Note: please note that this RV differs from the Zeta distribution.

To illustrate Whuber's comment
$$\begin{array}{c|ccccccccc}
& Y = 2 & Y =3& Y=4 & Y=5\\
\hline
 X =2  & \frac{1}{2^2} & \frac{1}{3^2} & \frac{1}{4^2} &\frac{1}{5^2} & \dots\\
 X =3  & \frac{1}{2^3} & \frac{1}{3^3} & \frac{1}{4^3} & \frac{1}{5^3} & \dots\\
 X =4  & \frac{1}{2^4}  &\frac{1}{3^4} &\frac{1}{4^4} & \frac{1}{5^4} &\dots\\
\vdots& \\\text{etc.}&
\end{array}$$
And those terms can be seen as the product of a product $P(X=x|Y=y)P(Y=y)$ with a shifted geometric distribution $$P(X=x|Y=y) =   
\left(\frac{1}{y}\right)^{x} y(y-1)$$ and some sort of variant of a Zipf distribution $$P(Y=y) =   
 \frac{1}{y(y-1)}$$
