Sum of $n^{-s}$ over squarefree numbers

Question

I read about

$$ \sum_{n\in\mathcal{Q}}n^{-s} = \prod_{p}(1+p^{-s}) $$

in a book. Who first discovered this equation? Did it first appear in a paper?

Here $\mathcal{Q}$ is the set of squarefree positive integers, and $p$ stands for a prime number.

Answer 1

Why do you think it makes a big difference to restrict the summation to square free numbers ? The Euler product (with $\Re(s)> 1$ replaced by integer $n \ge 2$) is in Euler 1748 introduction-to-analysis-of-the-infinite book I (english traduction)

The expansion of $\prod_p (1+p^{-s})$ is not discussed but there is the expansion of $\prod_p \frac1{1+p^{-s}}$ and a few $\prod_p \frac1{1-\chi(p)p^{-s}}$ (for integer $n\ge 1$)

Not sure if he recognized the general concept of multiplicative function, Dirichlet series and Dirichlet convolution, the pseudo-randomness of $\mu(n)$, the Dirichlet characters..

Anyway he gave some ways to construct plenty of interesting Euler products so it is natural to check what happens with $\prod_p (1+p^{-s})$ which is kinda left as an obvious exercice.