### Riemann Products

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

$\prod_p (1 - \frac{1}{p})(1 - \frac{1}{p^s}) = \prod_p [(1 - \frac{1}{p}) - \frac{1}{p^s}(1 - \frac{1}{p})]$

$= \prod_p (1 - \frac{1}{p})[1 - \sum_{p_1} \frac{1}{p_{1}^{s}} + \sum_{p_1,p_2} \frac{1}{(p_1 p_2)^s} - \cdots ]$

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

$\frac{1}{\zeta (s)} = 1 + \sum_k (-1)^k \sum_{p_1,p_2, \cdots, p_k} \frac{1}{(p_1p_2 \cdots p_k)^s}$

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

$\frac{1}{\zeta (s)} = \sum_{even} \frac{1}{n^s} - \sum_{odd} \frac{1}{n^s} = \sum_n \frac{\mu (n)}{n^s}$

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

$\prod_p (1 - \frac{1}{p})(1 - \frac{1}{p^s}) = \prod_p [(1 - \frac{1}{p}) - \frac{1}{p^s}(1 - \frac{1}{p})]$

$= \prod_p (1 - \frac{1}{p})[1 - \sum_{p_1} \frac{1}{p_{1}^{s}} + \sum_{p_1,p_2} \frac{1}{(p_1 p_2)^s} - \cdots ]$

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

$\frac{1}{\zeta (s)} = 1 + \sum_k (-1)^k \sum_{p_1,p_2, \cdots, p_k} \frac{1}{(p_1p_2 \cdots p_k)^s}$

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

$\frac{1}{\zeta (s)} = \sum_{even} \frac{1}{n^s} - \sum_{odd} \frac{1}{n^s} = \sum_n \frac{\mu (n)}{n^s}$

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

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