How to compute the following product:
$$\prod_{k=1}^\infty\left(1+\frac{(-1)^k}{p_k}\right)=\left(1-\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1-\frac{1}{5}\right)\left(1+\frac{1}{7}\right)\dots$$
where $p_k$ is the $k^\text{th}$ prime number?
My Observation, but NOT SURE:
Using $k=1$ to $k=18$, the product is almost $0.578282825\dots$ and that value is
$$\frac{\frac{\pi}{5}+\frac{\pi}{6}}{2}=\frac{11\pi}{60}\approx0.575958653\dots$$
So, is the product really tends to $\frac{11\pi}{60}$?
Any help/hint will be appreciated. THANKS!
Just to give a few numbers.
Let us consider the partial products $$P_n=\prod _{k=1}^{10^n} \left(1+\frac{(-1)^k}{p_k}\right)$$ Computed exactly, here are the results (in decimal values) $$\left( \begin{array}{cc} n & P_n \\ 1 & 0.5849897151924156997471733 \\ 2 & 0.5738797348557373537954953 \\ 3 & 0.5733032174061992771374417 \\ 4 & 0.5732660657801225326295239 \\ 5 & 0.5732633085315702175401998 \\ 6 & 0.5732630922693932644226413 \end{array} \right)$$