Riemann hypothesis

Ottobre 30th, 2021 | by Marcello Colozzo |

Riemann hypothesis,trivial zeros,proof,non-trivial zeros
Fig. 1

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)

by the following property (see Edwards)

Conversely, zeros with non-zero imaginary part are called non-trivial zeros.

Conjecture (Riemann Hypothesis)


From the functional equation and from the distribution property of trivial zeros, we have

So that non-trivial zeros are distributed in the strip

The set of the real part of the non-trivial zeros has x=0 as an accumulation point. On the other hand, the following symmetry property follows from the functional equation:

It follows

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