Using Dirichlet series (no complex analysis required)
Theorem. Suppose $f,g$ are arithmetic functions such that
converge absolutely at $s\in\mathbb C$. Let $h=f*g$ be their Dirichlet convolution. Then
(and the latter series converges absolutely at $s$).
Proof. Given absolute convergence, we can rearrange the terms. $\square$
With $f=\bf1$ and $g=\mu$ we have $f*g=1$ for $n=1$ and $0$ otherwise.
If $\sigma>1$, both $F(s)=\zeta(s)$ and $G(s)$ converge absolutely. So
In particular, it follows that $\zeta(s)\neq0$.