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I recently read that if the number of multiplicative partitions of $n$ is $a_n$, McMahon and Oppenheim observed that its Dirichlet series generating function $f(s)$ has the product representation

$$f(s)=\sum_{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}=\prod_{k=2}^{\infty }{\frac {1}{1-k^{-s}}}.$$

What is the proof for this? I didn't find any complete proof to the above correspondence. Does someone know where to find it? Or at least how it works?

To be clear, I'm interested not in the veracity of symbolic identity from an algebraic perspective; rather, I wonder why this generating function corresponds to the number of partitions for some number $n$.

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  • $\begingroup$ The reason is the same as Euler's product formula for Riemann's zeta function. Try to see why this is true by rewriting $\frac{1}{1-k^{-s}}$ as $1 + \frac{1}{k^s} + \frac{1}{k^{2s}} + \cdots$ and expand a few terms. $\endgroup$ – Hw Chu Apr 19 '18 at 18:32
  • $\begingroup$ @HwChu I understand the algebraic equality, but why does this correspond to the number of multiplicative partitions of $n$? $\endgroup$ – Tiwa Aina Apr 19 '18 at 19:32

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