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 I j and u = 0 is at its two ends. If | u ( t , β )| has only one peak on each I j , 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, u t ( t , β ) has opposite signs at two ends t = t j , t j+1 of I j , also does, then there exists some inner point t ′ such that v ( t′ , β ) = 0 . Therefore {| u |,| v |/ β } in I j form a peak-valley structure such that has positive lower bound independent of t ∈ I j ( i.e . RH holds in I j ). As u ( t , β ) does not have the finite condensation point (unless u = cons t . ), any finite t surely falls in some I j , 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.