Final Shape of Ztar's Steps 1

In Analytic Number Theory the prime numbers follow a logarithmic distribution and in my definition of complex space distances are logarithmic. Therefore the possibility of having complex waveforms following prime number patterns arises. As a consequence a waveform can represent a series of vectors so this also establishes the possibility of having complex forces and telling us something about their behaviour.
I also define the complex plane has having one face in 3D (the interior face) and having one face in 4D (the exterior face).

The Final Shape and the Dirichlet C Function
In figure “The Big Oh Notation” we have the limit that includes an arithmetic progression.

f(n) = 1 + 1/2 + 1/3 + ….. + 1/n

I will use this to define what should be the final shape of the ztar’s steps.
In the equation itself a difference of log n is included. I would suggest this determines the relative position upon the ztar’s arm. I would expect the numbers n to cover the prime numbers but this seems to deviate from Dirichlet’s equation. Hat deviation should not actually exist when n is prime we determine the site of a step (determined by the applied force and its wavelength) on the ztar and for all other n there is a position mid-phase.
 
In figure “Shape of Ztar's Steps 1” we have my original approximation. I did not see any way of reconciling number theory to prevent an abrupt change in the step for a ztar.

Considering Dirichlet’s equation it is not obvious but after much consideration I believe that a cumulative effect of a force along a ztar’s arm could be (and probably is) correct.

In figure “Shape of Ztar's Steps 2” we have Dirchlet’s equation applied to the steps providing a gradual decay to follow the asymptotic arm of the ztar.

In figure “Shape of Ztar's Steps 3” I have tried to improve upon my first attempt but I am still dealing with primes. Although the next figure is more accurate, I anticipate that this approach may be applicable if more that one force is applied along the ztar.

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