Number Theory

    

Geometric Interpretations of Riemann Hypothesis and the Proof

Authors: Tae Beom Lee

The Riemann zeta function(RZF), ζ(s), is a function of a complex variable s=x+iy. Riemann hypothesis(RH) states that all the non-trivial zeros of RZF lie on the critical line, x=1/2. The symmetricity of RZF zeros implies that if ζ(α+ iβ)=0, then ζ(1-α+ iβ)=0, too. In geometric view, if RH is false, two trajectories ζ(α+ iy) and ζ(1-α+ iy) must intersect at the origin when y=β. But, according to the functional equations of RZF, two trajectories ζ(α+ iy) and ζ(1-α+ iy) can’t intersect except when α=1/2. So, they can’t intersect at the origin, too, proving RH is true.

Comments: 5 Pages.

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[v1] 2024-03-07 07:52:34

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