That $1/\zeta(s)$ is absolutely and uniformly convergent and therefore analytic for $\Re(s)>1$, and that it has meromorphic continuation to the whole complex plane, is elementary. But the issue of convergence on $1/2\leq\Re(s)\leq 1$ is more interesting.

• at $\Re(s)=1/2$ it diverges

Lucia gave a very nice proof of this fact here on MO.

• at $\Re(s)=1$ it converges

This follows with a bit of work from the estimate

$$\sum_{n<x}\frac{\mu(n)}{n^s}=\frac{1}{2\pi i}\int_{c-it}^{c+it}\frac{1}{\zeta(s+w)}\frac{x^w}{w}dw+O\left(\frac{x^c}{Tc}\right)+O\left(\frac{\log x}{T}\right)$$

which is proved in more generality in section 3.12 of Titchmarsh's book on Riemann zeta function theory. The application to $a_n=\mu(n)$ is worked out in section 3.13.

• at $1/2<\Re(s)< 1$ it converges if and only if the Riemann hypothesis holds

The only if part is obvious. The converse follows from the same kind of estimate as above and the residue theorem. See section 14.25 of Titchmarsh.

That $1/\zeta(s)$ is absolutely and uniformly convergent and therefore analytic for $\Re(s)>1$, and that it has meromorphic continuation to the whole complex plane, is elementary. But the issue of convergence on $1/2\leq\Re(s)\leq 1$ is more interesting.

• at $\Re(s)=1/2$ it diverges

Lucia gave a very nice proof of this fact here on MO.

• at $\Re(s)=1$ it converges

This follows with a bit of work from the estimate

$$\sum_{n<x}\frac{\mu(n)}{n^s}=\frac{1}{2\pi i}\int_{c-it}^{c+it}\frac{1}{\zeta(s+w)}\frac{x^w}{w}dw+O\left(\frac{x^c}{Tc}\right)+O\left(\frac{\log x}{T}\right)$$

which is proved in more generality in section 3.12 of Titchmarsh's book on Riemann zeta function theory. The application to $a_n=\mu(n)$ is worked out in section 3.13.

• at $1/2<\Re(s)< 1$ it converges if and only if the Riemann hypothesis holds

The only if part is obvious. The converse follows from the same kind of estimate as above and the residue theorem. See section 14.25 of Titchmarsh.