# A Dynamical Approach to Riemann Zeta Function

Here is a definition of the Riemann zeta function.

Def 1.1: The Riemann zeta function $\zeta(s)$ is defined as

$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.$

The function is defined on the whole complex plane but $s = 1$. Riemann proved that the zeta function has a symmetric property, which is

$\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$   (1)

Originally, $\zeta$ is defined on a complex plane to itself, but for now, let us limit the domain and the range of the function to real numbers in order to create a dynamical system. The first task is to look for fixed points of $\zeta$.

Thm 1.1: There are infinitely many fixed points of $\zeta$ on $s < 1$.

Pf. of Thm 1.1

The functional equation (1) can be re-written into

$\zeta(1-s) = \pi^{-1}(2\pi)^{1-s} \Gamma(s) \cos\left(\frac{\pi s}{2}\right) \zeta (s)$   (2)

The Taylor series of $e^z$ is

$e^z = \sum_{k=0}^\infty \frac{(z)^k}{k!}$

Since $e^z$ converges on the entire complex plane, the sequence of the terms $\frac{(2\pi)^k}{k!}$ converges to zero. Hence,

$\text{As } s \in \mathbb{N}, s \to \infty, \hspace{0.5cm} \frac{(s-1)!}{(2\pi)^{s-1}} = \Gamma(s) (2\pi)^{1-s} \to \infty$

Since $\zeta(s)$ asymptotically approaches to 1, $2(2\pi)^{-s}\Gamma(s)\zeta(s)$ still diverges. However, $\cos\left(\frac{\pi s}{2}\right)$ forces the whole equation to return to zero for every 2 units. Therefore, it can be concluded that $y = \zeta(1-s)$ crosses $y = s$ infinitely many times. Thus, there are infinitely many fixed points along $s < 1$.

As the graph above suggests, there are infinitely many periodic points along the line $s < -20$, and there are two other fixed points which are $s = -.295905005575214...$ and $s = 1.83377265168027...$ numerically. Before we go further, here I would like to introduce a pair of terminologies;

Def. 1.2: Let $p$ be a periodic point of period $n$ under a function $f$.

If $|(f^n)'(p)| < 1$, then $p$ is an attracting periodic point (an attractor).
If $|(f^n)'(p)| > 1$, then $p$ is a repelling periodic point (a repellor).

And, here is a proposition regarding the newly introduced terminologies.

Prop. 1.1: Let $p$ be an attractor. Then, there is an open interval $U$ about $p$ such that if $x \in U$, then

$\lim_{n\to \infty} f^n(x) = p.$ [1]

Prop. 1.2: Let $p$ be a repellor. Then, there is an open interval $U$ of $p$ such that, if $x \in U\setminus \{p\}$, then there exists $k > 0$ such that $f^k(x) \not \in U$. [2]

All the periodic points along $s<-20$ and $s=1.834...$ are repellors. Plus, all the points on $s > 1$ besides the periodic point are attracted to a cycle of prime period 2, which is $(1, \infty)$.

The only fixed point left, $s = -.296...$ , is the only attractor among the fixed points. If $s=0$ was entered in the dynamical system $\zeta^n(s)$ for an arbitarily large $n$ converged to $s = -.296...$; thus I hoped to find the distribution of $s = -.296...$ along $\zeta^n: \mathbb{C} \to \mathbb{C}$ for arbitrarily large $n$ in order to observe the distribution of zeros of Riemann zeta function. However, this approach bears two crucial flaws.

The first flaw is that $s=-.296...$ is indeed a strong attractor that sucks up all the real numbers on $s < 1$ to itself after sufficiently many re-iterations of $\zeta(s)$; in other words, non-trivial zeros are not the only $s \in \mathbb{C}$ which converge to the constant.

I assumed that my plan would be somehow feasible if I could define a $\zeta$ on dynamical system from a set $A \subset \mathbb{C}$ to itself. $\zeta: A \to A$. However, we need $\forall s \in A, \Re(s) \geq 1/2$, yet a non-trivial zero will send $\zeta (s) = 0 \not \in A$; thus it is impossible to find such $A$.

Quite disappointingly yet expectedly, the dynamical approach seems to be pretty futile in figuring out the trivial zeros of $\zeta$.

1. Devaney, Robert L. An Introduction to Chaotic Dynamical Systems, 2nd Edition: Westview, 2003. Print.
2. IBID.