Geometric Proof of Riemann Conjecture

This paper proves Riemann conjecture (RH), i.e., that all the zeros in critical region of Riemann ξ -function lie on symmetric line 1 2 σ= . Its proof is based on two important properties: the symmetry and alternative oscillation for u iv ξ= + . Denote it . Riemann proved that u is real and 0 v ≡ for 0 β= (the symmetry). We prove that the zeros of u and v for 0 β> are alternative, so ( ) ,0 u t is the single peak. A geometric model was proposed. 1 is called the root-interval of , u t , if is inside j I and 0 u = is at its two ends. If ( ) , u t β has only one peak on each j I , 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 0 β> were derived. By peak ( ) , u t β will develop toward its convex direction. Besides, u has opposite signs at two ends β also does, then there exists some inner point t ′ such that ( ) , 0 v t β ′ = . Therefore { } , u v β in j I form a peak-valley structure such that ( ) 0 j u v ξ β µ β + ≥ > has positive lower bound independent of j t I ∈ (i.e. RH holds in j I ). As ( ) , u t β does not have the finite condensation point (unless . u const = ), any finite t surely falls in some j I , then 0 ξ > 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.

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. Hilbert's statement. J. Conrey [4] pointed out that "It is my belief, RH is a genuinely arithmetic question that likely will not succumb to methods of analysis". E. Bombieri [3] expected that "For them, we do not have algebraic and geometric models to guide our thinking, and entirely new ideas may be needed to study these intriguing objects". These advices make us realize that the analysis of the infinite series is hopeless and we should pay more attention to the algebraic and geometric analysis. We have also noted a new trend to give up ζ and turn to ξ . P. Sarnak [7] (2004) pointed out that "Riemann showed how to continue zeta analytically in s and he established the Functional Equation:

Introduction
Γ being the Gamma function. RH is the assertion that all the zeros of ( ) which is the first paper to study the equivalence of Ξ and RH. He computed Ξ and proposed a guess: if any part summation has the monotone zeros, then RH holds. He thought that the study of the Ξ -function was the right approach to RH. This is very important.
We have computed the Riemann ξ -function and other continuations of Euler ζ -function, and found that only ξ has the symmetry and alternative oscillation, which intuitively implies RH. Whereas others ζ have no the properties, and proving RH is hopeless.
We reread the original paper of Riemann (see [5]) and found his thought to study ξ -function. We list 4 terms concerning RH and the important progression as follows.  is Jacobi's function.
2) Introduced an entire function (which is a symmetrization) In critical domain ζ and ξ have the same zeros. Taking cos ln d , Im 0. d 2 In present point of view, using translating 1 2 s it should directly get Riemann's general formula [5] ( ) On critical line 3) Riemann said, "The number of roots of ( ) 0 t ξ = whose real parts lie between 0 and T is about 1905), and pointed out that, "One finds in fact about this many real roots within these bounds and it is very likely that all of the roots are real. One would of course like to have a rigorous proof of this" (i.e. RH).

4) He guessed a multiplication formula of ξ (it is proved by Hadamard
From these we see that Riemann had emphasized ξ , rather than ζ . (1932) found a formula unpublished in Riemann's manuscript(now called R-S formula, which was derived by ξ , and is large scale computing formula on critical line up to now), and Riemann had already computed the first several roots (due to the inspiration of R-S formula, we shall propose a new computing formula in next paper). 6) Lagarias [9] (1999) found the positivity 0

5) Siegel
property undiscussed by Riemann), which is the most essential progression since 1932, also the first equivalence to RH for ξ . Its proof requires the properties 3) and 4). s x  Definition 2 (single peak). If u has only one peak in each root-interval j I , called single peak, unless called multiple peaks(we shall prove no multiple peaks in theorem 3).

A Geometric Model of ξ
Using Newton-Leibnitz formula, the symmetry ( ) ( )   Above three cases prove (2.5). Similarly discuss the curve CEF.  Remark 2. We in [11] intuitively thought that ( ) , 0 t j u t β > will imply (2.5), this is not strict. Now (2.5) is strictly proved by three segments of convexity.

Geometric Proof of Riemann Conjecture
We shall regard which are single zeros(the double zeros are admitted). For any 1 0, 2 Basic theorem. All zeros of Riemann ξ -function lie on critical line.
Its proof consists of three theorems as follows.    Proof. We know that analytic function ( ) ( ) ( )