Zeta function representation in terms of bosonic/fermionic degrees of freedom

Question

Consider Euler's product representation of the zeta function

$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{i=1}^\infty \frac{1}{1-p_i^{-s}},$$

with $\text{Re}(s)>1$, and $p_i$ is the $i^{th}$ prime number, i.e. $p_1=2,p_2=3,p_3=5,...$

Now, if we think of $s$ as the inverse (complex) temperature and $\epsilon_i=\log p_i$ as the energy of an exotic Bose particle in "quantum state" $i$. Then, the right hand side of the above equation could be considered as a partition function of an ideal gas of identical (exotic) Bose particles (see "Bose-Einstein statistics").

On the other hand, the product representation of the correponding (exotic) fermionic partition function, say $\zeta_F(s)$, is equal to (Fermi-Dirac statistics)

$$ \zeta_F(s)=\prod_{i=1}^\infty (1+p_i^{-s}).$$

1. What is the series representation of $\zeta_F(s)$ in analogy to $\zeta(s)$?

2. What is the corresponding analytical continuation for $\zeta_F(s)$?

3. Is this approach encouraging?