  We show that an obstruction of number-theoretical nature appears as a
necessary condition for the global hypoellipticity of the pseudo-differential
operator $L=D_t+(a+ib)(t)P(D_x)$ on $\mathbb{T}^1_t\times\mathbb{T}_x^{N}$.
This condition is also sufficient when the symbol $p(\xi)$ of $P(D_x)$ has at
most logarithmic growth. If $p(\xi)$ has super-logarithmic growth, we show that
the global hypoellipticity of $L$ depends on the change of sign of certain
interactions of the coefficients with the symbol $p(\xi).$ Moreover, the
interplay between the order of vanishing of coefficients with the order of
growth of $p(\xi)$ plays a crucial role in the global hypoellipticity of $L$.
We also describe completely the global hypoellipticity of $L$ in the case where
$P(D_x)$ is positively homogeneous. Additionally, we explore the influence of
irrational approximations of a real number in the global hypoellipticity.
