Traces of the Riemann zeta function on the complex plane.

v21 I have updated Appendix D. The manuscript proposes an alternative to the convergence points, considered useful results of the Riemann zeta function. Below I call a the real part of (s) and b the imaginary part. The formulation (s) =₍ ₁1nˢ is considered unsuitable, since it converges only if a>1. In the manuscript I highlight that the divergence correspond to a spiral and that the origin of the spiral, for the same (s), coincides with the point of convergence recognized as valid. I describe three methods for calculating the origin of the last diverging spiral resulting from (s) =₍ ₁1nˢ The first method uses the midpoint of one of the two vectors, which compete for the closest proximity to the origin. The accuracy provided by this method can be considered useful only if a>0 and (for a=1/2 must be at least) b>10000. The second method is definitely more precise, it is enough that b0 but it requires that a=1/2. In appendix (D) I describe a method that improves with evidence (but I don’t consider it definitive), the accuracy of the first method. Also the third method works for any value of a, but the precision is only useful if a>0. Concluding. I studied the traces resulting from three formulations of the Riemann zeta function. In the manuscript I describe the reasons why (I maintain that) the Riemann hypothesis is true and (I am convinced that) the formulation (s) =₍ ₁1nˢ is the best possible.