Convergence or divergence of $\sum\limits_n(-1)^{\pi(n)}\frac1n$ where $\pi(n)$ is the number of primes less than or equal to $n$ Consider $$\sum_{n=1}^{\infty}\frac{(-1)^{\pi(n)}}{n}$$ where $\pi(n)$ is the number of primes less than or equal to $n$.
Does this sum converge or does it diverge? Are there any results related to this?  
 A: Let P(i) denotes the i-th prime number, then when n is between P(i) and P(i+1)-1, $\pi$(n) doesn't change.
Group these terms with the same sign together, we obtain a new series: $\sum \hat{a}(k)$. Now we have an alternating series. we use the Leibniz's theorem:
"If the absolute value of  $\hat{a}(k)$ decreases with k, and $\lim_{n\to\infty}\hat{a}(k)=0$ then, this series is convergent."
But now we should prove that:
$$
\hat{a}(k)=\frac{\Delta P_k}{P_k}\to 0
$$
where $\Delta P_i$ is the number of integers between the k-th prime and (k+1)-th prime minus 1.  i.e. the 'gap' between two consecutive prime numbers.
once we finish to do that, we can say the original series is also convergent, because the partial sum of the original series is between  the sums of the new series:
$$\hat{S}(k)\le S(n_k\le i\le(n_{k+1}))\le \hat{S}(k+1)$$
(since the terms between have the same sign, and thus the sum is monotonic in between.)
But I fail to prove 
$$
\hat{a}(k)=\frac{\Delta P_k}{P_k}\to 0
$$
Are there any primes between $P_k$ and $2P_k$?
if not, $\Delta P_k> P_k$. and the above limit is wrong.
A: It is not currently known whether or not the sum $\sum_{n=1}^\infty \frac{(-1)^{\pi(n)}}{n}$ converges.
I've been working on a related problem that sheds some light:
Erdos asks if the sum $\sum_{n=1}^\infty \frac{ (-1)^n n}{p_n}$ converges.  Here $p_n$ is the n-th prime number.  By using the prime number theorem and estimating the difference between $\frac{n}{p_n}$ and $\frac{n+1}{p_{n+1}}$ for odd and even $n$, one can show that Erdos' series converges simultaneously with the series $\sum_{n=2}^\infty \frac{(-1)^{\pi(n)}}{n \log n}$.  Here is a link to the problem I am working on https://mathoverflow.net/q/313999/8435
