Does $\sum _{n=1}^\infty\frac{\left(-1\right)^{\pi \left(n\right)}}{n}$ converge, where $\pi(x)$ is the prime counting function?
Convergence or divergence of $\sum\limits_n(-1)^{\pi(n)}\frac1n$ where $\pi(n)$ is the number of primes less than or equal to $n$