Geometric Proof of Riemann Conjecture

Abstract

This paper proves Riemann conjecture (RH), i.e., that all the zeros in critical region of Riemann ξ -function lie on symmetric line σ =1/2 . Its proof is based on two important properties: the symmetry and alternative oscillation for ξ = u + iv . Denote . Riemann proved that u is real and v ≡ 0 for β =0 (the symmetry). We prove that the zeros of u and v for β > 0 are alternative, so u (t,0) is the single peak. A geometric model was proposed. is called the root-interval of u (t,β) , if |u| > 0 is inside Ij and u = 0 is at its two ends. If |u (t,β)| has only one peak on each Ij, which is called the single peak, else called multiple peaks (it will be proved that the multiple peaks do not exist). The important expressions of u and v for β > 0 were derived. By , the peak u (t,β) will develop toward its convex direction. Besides, ut (t,β) has opposite signs at two ends t = tj , tj+1 of Ij , also does, then there exists some inner point t′ such that v (t′,β) = 0. Therefore {|u|,|v|/β} in Ij form a peak-valley structure such that has positive lower bound independent of t ∈ Ij (i.e. RH holds in Ij ). As u (t,β) does not have the finite condensation point (unless u = const.), any finite t surely falls in some Ij , then holds for any t (RH is proved). Our previous paper “Local geometric proof of Riemann conjecture” (APM, V.10:8, 2020) has two defects, this paper has amended these defects and given a complete proof of RH.