Prime orbits for some smooth flows on $\mathbb{T}^2$

We consider a class of smooth mixing flows $T^{α,γ}$ on $\mathbb{T}^2$ with one degenerated fixed point $x_0\in \mathbb{T}^2$ of power type $γ\in (-1,0)$. We prove that for a $G_δ$ dense set of $α\in \mathbb{T}$, a prime number theorem for $T^{α,γ}$ 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 $\log^{-A}N$).

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