We know that the Riemann zeta function is defined as
$$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},$$
for all $\Re(s)>1$.
Because of Euler product formula we also know that
$$\zeta(s) = \sum_{n=1}^\infty\frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}},$$ for all $\Re(s)>1.$
There are a lot of functions related to Riemann zeta function. For example
- $\zeta(s)=\zeta(s,1)$ where $\zeta(s,q)$ is the Hurwitz zeta function.
- $\zeta(s)=\operatorname{Li}_s(1)$, where $\operatorname{Li}_s(z)$ is the polylogarithm.
- $\zeta(s)=(1-2^{-s})^{-1}\chi_s(1)$, where $\chi_s(z)$ Legendre chi function
- $\zeta(s)=\Phi (1,s,1)$, where $\Phi(z, s, \alpha)$ is the Lerch zeta function
- $ \zeta(s) = (1-2^{1-s})^{-1}\eta(s)$, where $\eta(s)$ is the Dirichlet eta function
- $\dots$ and there are lot of other related functions such as multiple zeta function, Barnes zeta function, the Clausen function, etc.
Question. Are there Euler product formula type statements to other special functions?