  We consider a class of smooth mixing flows $T^{\alpha,\gamma}$ on
$\mathbb{T}^2$ with one degenerated fixed point $x_0\in \mathbb{T}^2$ of power
type $\gamma\in (-1,0)$. We prove that for a $G_\delta$ dense set of $\alpha\in
\mathbb{T}$, a prime number theorem for $T^{\alpha,\gamma}$ holds along a full
upper density subsequence. In particular it follows that for every $x\in
\mathbb{T}^2\setminus\{x_0\}$, the prime orbit $\mathbb{T}^2$.
  We also show that there exists a class of smooth weakly mixing flows on
$\mathbb{T}^2$ for which a prime number theorem holds. In fact we show that
there exists a dense set of smooth functions (in the uniform topology) for
which prime number theorem holds quantitatively (with an error term
$\log^{-A}N$).
