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Let $\zeta$ be the classical Riemann zeta function.

We define a differential equation on $\mathbb{R}^{2} \setminus \{1\}$ by $\dot Z= \zeta(Z)$. From a foliation point of view this vector field can be counted as a smooth vector field on whole $\mathbb{R}^{2}$ with the following equivalent formulation(They have the same trajectories).

$$\dot Z= \parallel z-1\parallel^2 \zeta(Z)$$

Then the field has a saddle point at $1$.

Are there some researches about this dynamical system?Are there closed orbits for this equation?The latter is equivalent to ask: "Are there zeroes of the Riemann Zeta function whose Taylor expansion (after translation to the origin and real rescalling ) is in the form $"iz+...."$. Every zero of a holomorphic map with this linear part is necessarily a center, a singularity surrounded by a band of closed orbits.

Let $\zeta$ be the classical Riemann zeta function.

We define a differential equation on $\mathbb{R}^{2} \setminus \{1\}$ by $\dot Z= \zeta(Z)$. From a foliation point of view this vector field can be counted as a smooth vector field on whole $\mathbb{R}^{2}$ with the following equivalent formulation(They have the same trajectories).

$$\dot Z= \lVert z-1\rVert^2 \zeta(Z)$$

Then the field has a saddle point at $1$.

Are there some researches about this dynamical system?Are there closed orbits for this equation?The latter is equivalent to ask: "Are there zeroes of the Riemann Zeta function whose Taylor expansion (after translation to the origin and real rescalling ) is in the form $"iz+...."$. Every zero of a holomorphic map with this linear part is necessarily a center, a singularity surrounded by a band of closed orbits.

Let $\zeta$ be the classical Riemann zeta function.

We define a differential equation on $\mathbb{R}^{2} \setminus \{1\}$ by $\dot Z= \zeta(Z)$. From a foliation point of view this vector field can be counted as a smooth vector field on whole $\mathbb{R}^{2}$ with the following equivalent formulation(They have the same trajectories).

$$\dot Z= \parallel z-1\parallel^2 \zeta(Z)$$

Then the field has a saddle point at $1$.

Are there some researches about this dynamical system?Are there closed orbits for this equation?

Let $\zeta$ be the classical Riemann zeta function.

We define a differential equation on $\mathbb{R}^{2} \setminus \{1\}$ by $\dot Z= \zeta(Z)$. From a foliation point of view this vector field can be counted as a smooth vector field on whole $\mathbb{R}^{2}$ with the following equivalent formulation(They have the same trajectories).

$$\dot Z= \parallel z-1\parallel^2 \zeta(Z)$$

Then the field has a saddle point at $1$.

Are there some researches about this dynamical system?Are there closed orbits for this equation?The latter is equivalent to ask: "Are there zeroes of the Riemann Zeta function whose Taylor expansion (after translation to the origin and real rescalling ) is in the form $"iz+...."$. Every zero of a holomorphic map with this linear part is necessarily a center, a singularity surrounded by a band of closed orbits.

Let $\zeta$ be the classical Riemann zeta function.

We define a differential equation on $\mathbb{R}^{2} \setminus \{1\}$ by $\dot Z= \zeta(Z)$.

Are there some researches about this dynamical system?Are there closed orbits for this equation?

Let $\zeta$ be the classical Riemann zeta function.

We define a differential equation on $\mathbb{R}^{2} \setminus \{1\}$ by $\dot Z= \zeta(Z)$. From a foliation point of view this vector field can be counted as a smooth vector field on whole $\mathbb{R}^{2}$ with the following equivalent formulation(They have the same trajectories).

$$\dot Z= \parallel z-1\parallel^2 \zeta(Z)$$

Then the field has a saddle point at $1$.

Are there some researches about this dynamical system?Are there closed orbits for this equation?