Riemann Products

Speaking of Euler products associated to the zeta function, the totient function leads one to consider the product

$\prod _p(1-\frac{1}{p})(1-\frac{1}{p^s})=\prod _p\left[\right(1-\frac{1}{p})-\frac{1}{p^s}(1-\frac{1}{p}\left)\right]$
$=\prod _p(1-\frac{1}{p})[1-\sum _{p_1}\frac{1}{p_1^s}+\sum _{p_1,p_2}\frac{1}{(p_1{p}_2{)}^s}-\cdots ]$

where one of the terms on the right hand side is a sum over $k$ distinct prime factors. That is, in cancelling the $\phi (\infty )$ factors, we have that

$\frac{1}{\zeta \left(s\right)}=1+\sum _k(-1{)}^k\sum _{p_1,p_2,\cdots ,p_k}\frac{1}{(p_1{p}_2\cdots p_k{)}^s}$

which is a simple sum over all ordinals $n$ composed of single prime factors. This may be rewritten

$\frac{1}{\zeta \left(s\right)}=\sum _{even}\frac{1}{n^s}-\sum _{odd}\frac{1}{n^s}=\sum _n\frac{\mu \left(n\right)}{n^s}$

where the parity counts the number of prime factors in $n$, and $1$ is an even prime. The function $\mu \left(n\right)$ is the Mobius function, which is zero for $n$ with repeated prime factors.