Is this class of infinite products over primes well known?
Investigating $\dfrac{\zeta(s-1)}{\zeta(s)}$ I was able to generate this series for $\pi$:
$\displaystyle\dfrac{\pi^2}{15}=\prod_{p\in\text{ prime}}\dfrac{p^2}{p^2+1}$
It's fairly straightforward to obtain and it doesn't converge particularly quickly. For example by the fourth term it gives:
$\pi\approx\sqrt{2205/208}\approx3.25$
I can't find a reference to this product anywhere aside from the obvious, and interesting similarity to Euler's:
$\displaystyle\dfrac{\pi^2}{6}=\prod_{p\in\text{ prime}}\dfrac{p^2}{p^2-1}$
Equating their $\pi$s yields the interesting product of primes:
$\displaystyle\prod_p\frac{p^2-1}{p^2+1}=\frac{2}{5}$
This has some curious properties:
$\displaystyle\int_{-1}^1 \frac{p^2-1}{p^2+1} dp = 2 - \pi$
In fact if we subtract the diverging parts:
$\displaystyle\int_{0}^{\infty} \left(\frac{p^2-1}{p^2+1}-1\right) dp = - \pi$
Although I've not done so yet, I think an infinite number of such products over primes can be obtained by this method; all of which equal a rational number. I simply used the quotients between $\zeta(2), \zeta(3), \zeta(4)$ in such a way that $\zeta(3)$ cancelled out.
I was curious as a next step to have a look at what the sequence of such products looks like, and what is its limiting behaviour as $s\to\infty$ which might tell us something about exact forms for $\zeta(2n+1)$. But I wondered if I'm obviously following in somebody else's footsteps here.
These are called Euler Products.
In particular, $$\frac{\pi^2}{15} = \frac{\zeta(4)}{\zeta(2)} = \prod_p \left( 1+p^{-2} \right)^{-1}.$$
See the examples here for more:
https://en.m.wikipedia.org/wiki/Euler_product
Edit: using the identities in that link, it can be seen that if the real part of $s$ is greater than 2 then we have $$\frac{\zeta(s-1)}{\zeta(s)} = \prod_p \frac{1-p^{-s}}{1-p^{1-s}}$$ and this is also equivalent to $$\sum_{n=1}^{\infty} \frac{\phi(n)}{n^s}.$$
I found the equivalency on page 231 of Apostol’s Introduction to Analytic Number Theory (pdf can be found here http://www.zuj.edu.jo/?wpdmdl=13005). There are a few identities there which are written as exercises or standard examples.