The Symmetry of Riemann ξ-Function

To prove RH, studying ζ and using pure analysis method likely are two kinds of the incorrect guide. Actually, a unique hope may study Riemann function ( ) u iv ξ τ = + , it τ β = + , 1 2 β σ = − by geometric analysis, which has the symmetry: 0 v = if 0 β = , and ( ) ( ) 0 , , d t v t u t r r β β = −∫ . Assume that u is single peak in each root-interval 1 , j j j I t t +   =   of u for any fixed ( ] 0,1 2 β ∈ , using the slope t u of the single peak, we prove that v has opposite signs at two end-points of j I , there surely is an inner point so that 0 v = , so { } , u v β form a local peak-valley structure, and have positive lower bound ( ) , 0 j u v t ξ β μ β = + ≥ > in j I . Because each t must lie in some j I , then 0 ξ > is valid for any t. In this way, the summation process of ξ is avoided. We have proved the main theorem: Assume that ( ) , u t β is single peak, then RH is valid for any [ ) 0, t∈ ∞ , ( ] 0,1 2 β ∈ . If using the equivalence of Lagarias (1999), the assumption of single peak can be canceled. Therefore our new thinking is that we have found the local peak-valley structure of ξ , which may be the geometry structure expected by Bombieri (2000), and proposed a basic framework of proving RH by geometric analysis.

β ∈ . If using the equivalence of Lagarias (1999), the assumption of single peak can be canceled. Therefore our new thinking is that we have found the local peak-valley struc-

Introduction Symmetry
We begin with two functions ζ and ξ introduced by Riemann. Euler (1737) proved the product formula over all prime numbers ( ) which is analytically extended over the whole complex plane, except for 0,1 s = .
Furthermore Riemann introduced an entire function and had gotten the second expression then proposed a proposition: Riemann Hypothesis (RH). In the critical region Why so difficult? We think there are two kinds of incorrect guide: studying ζ and using pure analysis method. We think that studying ζ is hopeless, and using pure analysis method has always met a wide gap: How to prove no zero for the infinite series? Conrey [5] pointed out that "It is my belief, RH is a genuinely arithmetic question that likely will not succumb to methods of analysis". Besides, Bombieri [3] (2000) pointed out 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". Thus I felt that a unique hope is to study ξ by geometric analysis.
Recall that Riemann had cleverly designed the function ξ in (1.4). He took This is the most important symmetry on critical line. But so far there are a few work on ξ , even ξ is denied. Denoting   , 0 u t β > inside j I . Definition 2. If u in each root-interval j I has only one peak, called the single peak, else called the multiple peaks (Actually the multiple peak case does not exist).
In numerical experiments, we found an important fact as follows. If using the equivalence of Lagarias [8], it is proved that the multiple peak case of ( ) , u t β does not exist, thus a complete proof of RH can be given [9]. If only using the expression (1.4) of ξ , the conclusion has not be proved yet. This is an unsolved problem.
Therefore the new thinking in this paper is that we have found the local peak-valley structure of ξ , which may be the geometry structure expected by Bombieri (2000), and proposed a basic framework of proving RH by concise geometric analysis.

Find Local Geometry Property by Computing
The norm ( ) It is known that ξ has the exponential decay [6] [9] ( ) To make the figures of { } , u v , we take a changing scale     "Give up method of analysis, directly study the geometry property of ξ itself".
So we came back to "positive phase-difference" once again, but now we find that it is a local peak-valley structure.
We explain the local peak-valley structure in Figure 2

Local Peak-Valley Structure in Single Peak Case
For fixed ( ] 0, 0.5 β ∈ the zeros j t of ( ) , u t β form an infinite sequence dependent on β We shall take them as the base and consider only single peak case.
The slope of single peak. For any 0 β ≥ , there are 0 t u > from negative peak to positive one, and 0 t u < from positive peak to negative one.
Besides, by basic expression we know that i.e., v β still is a valley curve and ( ) , 0 j t ξ µ β ≥ > in j I . Theorem 2 is proved.  From Theorem 2, we found a wonderful property: When 0 β ≥ increases, these root-intervals have a tendency to get more unform.
This property makes RH be still valid when β increases (e.g. RH is valid for 1 2 β =