Riemann hypothesis
Ottobre 30th, 2021 | by Marcello Colozzo |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)
Proof
From the functional equation and from the distribution property of trivial zeros, we have
So that non-trivial zeros are distributed in the stripThe 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
Tags: non-trivial zeros, proof, Riemann hypothesis, trivial zeros
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