Divergent Products over Prime Numbers

List of divergent infinite products over the set of prime numbers $\mathbb{P}$ .
In this context, $\zeta(1)$ maps to 1, not to $\gamma$ (the Euler-Mascheroni constant).

$\zeta(z)$ denotes the Riemann zeta function.

TRIBES

$\displaystyle \prod_{p\in\mathbb{P}} \, \sum_{l=0}^{m-1} \, \frac{1}{p^{lr}} = \frac{\zeta(r)}{\zeta(mr)}$

$\displaystyle \prod_{p\in\mathbb{P}} \, \sum_{l=0}^{m-1} \, p^{lr} = \frac{\zeta(r)}{\zeta(mr)} \, (2\pi)^{2(m-1)r}$

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