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

TL;DR

This paper proves prime number theorems for certain smooth mixing flows on the 2-torus with degenerate fixed points, demonstrating prime orbit distribution and establishing results for weakly mixing flows with quantitative error bounds.

Contribution

It introduces new prime number theorems for smooth flows on 2 with degenerate fixed points and constructs flows with quantitative prime orbit distribution results.

Findings

Prime number theorem holds along a full upper density subsequence for generic flows.

Existence of smooth weakly mixing flows satisfying prime number theorem.

Quantitative error bounds ( \,  \, ext{log}^{-A}N) for prime orbit distribution.

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