My goal is to show that $\varphi(s)$ is related to prime numbers.
Define $$ \varphi(s)=\sum_{n=1}^\infty \big(e^{-n^{-s}}-1\big). $$
Note that $e^x-1=\sum_{k\geq 1} x^k/k!$ for all $x$. One can use this expansion to express $\varphi(s)$ as a sum of Riemann zeta functions:
$$ \varphi(s)=\sum_{k=1}^\infty \frac{(-1)^k}{k!} \zeta(sk)$$
This means $\varphi(s)$ can be expressed as a sum of products of primes:
$$ \varphi(s)=\sum_{k=1}^\infty\frac{(-1)^k}{k!}\bigg(\prod_{\text{p prime}} \frac{1}{1-p^{-sk}{}}\bigg)$$
Is this correct?