Reverse of divergent Riemann zeta function. Try to play it.
Given for example the formulation ζ(s)=∑n≥1(1/n^s), at each increment of (n) a new point on the complex plane is defined. Starting from the origin of the complex plane, the ‘traces’ in question are formed of the vectors that connect (in sequence) these points.
Speaking of (s), I call (a) the real part and (b) the coefficient of the imaginary part, then s=a+b*i.
Provided that (b) is sufficiently large, the traces resulting from the zeta function of Riemann can be divided into three parts, below I call the first two useful, for a reason that I will explain later.
The first useful part of the traces tends to move away from the origin, it develops in a convoluted way reaching variable distances.
The second useful part of the traces, is characterized by the presence of particular polygonal spirals that follow one another, always better formed. Seen the similarity with clothoids I called them “pseudo-clothoids”.
The two useful parts behave like two arms of a mechanism that makes them both rotate, in case the value of (b) is changed in search of a zero. The second arm is the extension of the first, its rotation is in synchrony with that of the first arm, to which is added.
The rotation of the two arms cyclically brings the origin that concludes the second arm, to pass where the origin of the complex plane is located, but intercepts it only if the lengths of the two arms compensate.
In this manuscript I highlight that the Riemann hypothesis is true for the reason that, only if the real part of (s) is 1/2 it is possible to compensate the lengths, of the two useful parts of the trace.
The value of (b) is the engine of the rotations, only if a=1/2, the value of (b) results neutral with respect to the distances, between the two origins of the pseudo-clothoids.
Anyone interested in the subject can find two preprints on zenodo.org.
This is the link of the English version http://doi.org/10.5281/zenodo.8026759
This is the link of the original version in Italian http://doi.org/10.5281/zenodo.8026728