Riemann hypothesis

sabato, Ottobre 30th, 2021

As is well known the Riemann Zeta Function, is function of complex variable s=x+iy defined by the sum of the following Dirichlet series:

However, the function can be defined by holomorphic extension in all C excluding the polar singularity in s=1. More precisely, the holomorphic extension gives rise to the following functional equation:

From a property of the Dirichlet series that defines the zeta function, it follows the non-existence of zeros for Re(s) > 1, and from the functional equation it follows the non-existence of zeros with non-zero imaginary part for Re(s) < 0. On the other hand, for Re(s) < 0 there are zeros with a null imaginary part (trivial zeros). They are given by (fig. 1)