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

Abstract

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 ${\log }^{-A}N$).

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