Product of Zeta function

Question

Looking at this question on mathoverflow, I became curious about the value of

$$ p=\prod_{j=2}^\infty \zeta(j)^{-1} $$

In particular, is $p>0$ (assuming it converges)?

Answer 1

Since Euler's product ensures $\zeta(j)=\prod_{p}\left(1-\frac{1} {p^j}\right)^{-1}$ we have $$ \log\zeta(j)=\sum_{p}\sum_{m\geq 1}\frac{1}{m p^{mj}}, $$ $$ \sum_{j\geq 2}\log\zeta(j) = \sum_{p}\underbrace{\sum_{m\geq 1}\frac{1}{mp^m(p^m-1)}}_{\Theta\left(\frac{1}{p^2}\right)} $$ and the wanted product (which is the exponential of the opposite of the LHS) is clearly convergent.
Its numerical value is approximately $0.435757$.