Does anyone know what value the sum of squares of inverse of primes is equal to in terms of other known mathematical function? That is:
$$ \sum_{p\in \mathbb{P}} \frac{1}{p^2} $$
where $\mathbb{P}$ is the set of primes. This sum definitely converges by comparison to $1/n^2$ but I was wondering if it was an important constant and/or the value of a specific notable function.
Such series is close to $\log\sqrt{\frac{5}{2}}$, since $\frac{1}{p^2}$ is close to $\frac{1}{2}\log\frac{1+\frac{1}{p^2}}{1-\frac{1}{p^2}}=\text{arctanh}\frac{1}{p^2}$ and by Euler's product
$$ \sum_{p\in\mathcal{P}}\log\left(1+\frac{1}{p^2}\right) = \log\frac{\zeta(2)}{\zeta(4)},\qquad \sum_{p\in\mathcal{P}}-\log\left(1-\frac{1}{p^2}\right)=\log\zeta(2)$$ hence $$\sum_{p\in\mathcal{P}}\frac{1}{p^2}\approx \frac{1}{2}\log\frac{\zeta(2)^2}{\zeta(4)} = \log\sqrt{\frac{5}{2}}.$$ I leave to you to refine such (already quite accurate) approximation through a similar idea.