TITLE: A conjectural infinite series for $\frac{\pi}{2}$
QUESTION [6 upvotes]: Can you provide a proof for the claim given below?
In this Wikipedia article the constant $\pi$ is represented by the following infinite series:
$$\pi=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{\epsilon(n)}}{n}$$ where $\epsilon(n)$ is the number of prime factors of the form $p \equiv 1 \pmod{4}$ of $n$ . (Euler, 1748)
Similarly, we can formulate the following claim:
$$\frac{\pi}{2}=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{\kappa(n)}}{n}$$ where $\kappa(n)$ is the number of prime factors of the form $p \equiv 3 \pmod{4}$ of $n$ .
The SageMath cell that demonstrates this claim can be found here.
Now, I don't know how to start the proof. Any hints or references are welcomed.

REPLY [2 votes]: I believe A034947: Jacobi (or Kronecker) symbol (-1/n).  provides some insight where $\beta(s)$ is the Dirichlet beta function:
$$(-1)^{\kappa(n)}=A034947(n)\tag{1}$$
$$\frac{\beta(s)}{1-2^{-s}}=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N\frac{(-1)^{\kappa(n)}}{n^s}\right),\quad\Re(s)>0\tag{2}$$
$$\underset{s\to 1}{\text{lim}}\left(\frac{\beta(s)}{1-2^{-s}}\right)=\frac{\pi }{2}\tag{3}$$