[tex]f_X(x)=\begin{cases}\dfrac c{k^2}&\text{for }k\in\{1,2,\ldots\}\\0&\text{otherwise}\end{cases}[/tex]
Because this is a valid PMF, you must have
[tex]\displaystyle\sum_{k\ge1}f_X(x)=\sum_{k\ge1}\frac c{k^2}=1[/tex]
Recall that
[tex]\displaystyle\sum_{k\ge1}\frac1{k^2}=\dfrac{\pi^2}6[/tex]
so the above sum is
[tex]\displaystyle\sum_{k\ge1}\frac c{k^2}=\frac{c\pi^2}6=1[/tex]
which means that you must have
[tex]c=\dfrac6{\pi^2}\approx0.6079[/tex]
