Trace of the Riemann zeta function on the complex plane.

Given a formulation of Riemann zeta function, at each increment of (n) a point on the complex plane is defined, the traces in question are formed of the vectors connecting this points.[br][br]The traces resulting on the complex plane from the zeta function of Riemann, can be divided into two parts.[br]In the first part of the traces it is essential to focus attention on the start and end points of the vectors; in the second part it is essential to focus attention on the two origins of particular polygonal spirals, which I call "pseudo-clothoid".[br]All traces start from the origin of the complex plane, the first part of the traces tends to move away from the origin; it develops in a convoluted way reaching variable distances.[br]The two parts behave like two arms, of a mechanism which makes them both rotate but independently and clockwise; the hinge from which the rotation of the second arm begins, is located at the junction point with the first.[br]The rotation of the two arms cyclically brings the free end of the second arm, to pass where the origin of the complex plane is located; but only under one condition does it intercept it.[br]The condition is that the two parts of the trace must compensate each other; in this article I highlight how the compensation takes place between the two parts of the trace.[br]Given a complex number (s) I call (a) the real part and (b) the coefficient of the imaginary part; then s=a+b*i.[br]The value of (b) is the engine of the rotations; only if a=1/2, the value of (b) is neutral with respect to the distances between the two origins, of the pseudo-clothoids.[br]Anyone interested in the subject can find two preprints on zenodo.org.[br][br]This is the link of the English version http://doi.org/10.5281/zenodo.8026759[br]This is the link of the original version in Italian http://doi.org/10.5281/zenodo.8026728

Information: Trace of the Riemann zeta function on the complex plane.