Is $\sum_{n \geq 2} \frac{1}{\pi (n^2)}$ convergent or divergent?

Question

I wonder if $$\sum_{n \geq 2} \frac{1}{\pi (n^2)},$$ where $\pi(\cdot)$ is the prime-counting function, is convergent or not.

Please help me solve and understand this problem. Is related to analytic number theory.

Answer 1

By Chebyshev's theorem we have $\pi(n)\gg\frac{n}{\log n}$, hence the series is bounded by some constant times: $$\sum_{n\geq 1}\frac{2\log n}{n^2}$$ that is convergent.