An open problem in analytic number theory is to determine whether the sum
$\sum_n \frac{(-1)^n n}{p_n}$ converges. Here, $p_n$ is the n-th prime number.
It is known that this sum converges if and only if the series
$\sum_{m=2}^\infty \frac{(-1)^{\pi(m)}}{m \log m}$ converges.
Here, $\pi(n)$ denotes the prime counting function.
Can anyone use the prime number theorem to prove that these two series converge simultaneously?