Is Euler product formula equivalent to fundamental theorem of arithmetic (unique factorization theorem)?

Question

Is Euler product formula equivalent to fundamental theorem of arithmetic (unique factorization theorem) ?

$$\sum_{n=1}^\infty\frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}$$

We know that we can derive Euler product formula from fundamental theorem of arithmetic. Is the opposite also true ? Can we use Euler product formula to derive fundamental theorem of arithmetic.

The reason we are interested in this question is, if we can derive fundamental theorem of arithmetic from Euler product formula, does does this mean that Riemann Zeta function is a "complex version" of fundamental theorem of arithmetic ?

Answer 1

The answer is affirmative. Since $$\frac{1}{1-p^{-s}}=1+\frac{1}{p^s}+\frac{1}{p^{2s}}+\ldots, $$ we have the identity: $$\prod_{p}\frac{1}{1-p^{-s}}=\sum_{n\geq 1}\frac{h(n)}{n^s}$$ where $h(n)\in\mathbb{N}_{\geq 1}$ counts the number of ways of writing $n$ as a product of powers of different primes. Assuming $h(n)>1$ for some $n$, the identity $$\zeta(s)=\sum_{n\geq 1}\frac{h(n)}{n^s}$$ cannot hold for every $s\in\mathbb{R}_{>1}$. This proves the fundamental theorem of arithmetic.