  In this paper we consider the following Cauchy problem for the semi-linear
wave equation with scale-invariant dissipation and mass and power
non-linearity: \begin{align}\label{CP abstract} \begin{cases} u_{tt}-\Delta
u+\dfrac{\mu_1}{1+t} u_t+\dfrac{\mu_2^2}{(1+t)^2}u=|u|^p, \\ u(0,x)=u_0(x),
\,\, u_t(0,x)=u_1(x), \end{cases}\tag{$\star$} \end{align} where $\mu_1,
\mu_2^2$ are nonnegative constants and $p>1$. On the one hand we will prove a
global (in time) existence result for \eqref{CP abstract} under suitable
assumptions on the coefficients $\mu_1, \mu_2^2$ of the damping and the mass
term and on the exponent $p$, assuming the smallness of data in exponentially
weighted energy spaces. On the other hand a blow-up result for \eqref{CP
abstract} is proved for values of $p$ below a certain threshold, provided that
the data satisfy some integral sign conditions. Combining these results we find
the critical exponent for \eqref{CP abstract} in all space dimensions under
certain assumptions on $\mu_1$ and $\mu_2^2$. Moreover, since the global
existence result is based on a contradiction argument, it will be shown firstly
a local (in time) existence result.
