Here is a definition of the Riemann zeta function.
The function is defined on the whole complex plane but . Riemann proved that the zeta function has a symmetric property, which is
Originally, 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 .
Pf. of Thm 1.1
The functional equation (1) can be re-written into
The Taylor series of is
Since converges on the entire complex plane, the sequence of the terms converges to zero. Hence,
Since asymptotically approaches to 1, still diverges. However, forces the whole equation to return to zero for every 2 units. Therefore, it can be concluded that crosses infinitely many times. Thus, there are infinitely many fixed points along .
As the graph above suggests, there are infinitely many periodic points along the line , and there are two other fixed points which are and numerically. Before we go further, here I would like to introduce a pair of terminologies;
And, here is a proposition regarding the newly introduced terminologies.
All the periodic points along and are repellors. Plus, all the points on besides the periodic point are attracted to a cycle of prime period 2, which is .
The only fixed point left, , is the only attractor among the fixed points. If was entered in the dynamical system for an arbitarily large converged to ; thus I hoped to find the distribution of along for arbitrarily large in order to observe the distribution of zeros of Riemann zeta function. However, this approach bears two crucial flaws.
The first flaw is that is indeed a strong attractor that sucks up all the real numbers on to itself after sufficiently many re-iterations of ; in other words, non-trivial zeros are not the only which converge to the constant.
I assumed that my plan would be somehow feasible if I could define a on dynamical system from a set to itself. . However, we need , yet a non-trivial zero will send ; thus it is impossible to find such .
Quite disappointingly yet expectedly, the dynamical approach seems to be pretty futile in figuring out the trivial zeros of .