1. Introduction Symmetry
We begin with two functions and introduced by Riemann. Euler (1737) proved the product formula over all prime numbers
which converges for , but diverges for . Taking the complex variable , B. Riemann (1859) had gotten the first expression
which is analytically extended over the whole complex plane, except for .
Furthermore Riemann introduced an entire function
and had gotten the second expression
then proposed a proposition:
Riemann Hypothesis (RH). In the critical region , all the zeros of lie on the critical line , which is called the non-trivial zeros.
A lot of numerical experiments, for example  , verified that RH is valid. However, RH has not been proved to be valid or false in theory     . Why so difficult? We think there are two kinds of incorrect guide: studying and using pure analysis method. We think that studying is hopeless, and using pure analysis method has always met a wide gap: How to prove no zero for the infinite series? Conrey  pointed out that “It is my belief, RH is a genuinely arithmetic question that likely will not succumb to methods of analysis”. Besides, Bombieri  (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 by geometric analysis.
Recall that Riemann had cleverly designed the function in (1.4). He took to get the real expression
This is the most important symmetry on critical line. But so far there are a few work on , even is denied. Denoting , and , by the symmetry we have
Lemma 1 (Basic expression). For any and , the imaginary part v can be uniquely expressed by in the form
Proof. Using , and C-R condition , we get (1.6), which plays an important role in studying .
Corollary 1. is uniformly bounded with respect to .
Definition 1. For any fixed , the sub-interval is called the root-interval, if the real part , and inside .
Definition 2. If in each root-interval 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.
Proposition 1. For any fixed and in each root-interval , assume that has opposite signs at and , and at some inner point, then form local peak-valley structure, and norm in , i.e. RH is valid in .
Using the symmetry of and the slope of the single peak, we have proved the assumption of the proposition 1. Because each t must lie in some , thus is valid for any t. It proves
The main theorem. Assume that is the single peak, then RH is valid for any .
If using the equivalence of Lagarias , it is proved that the multiple peak case of does not exist, thus a complete proof of RH can be given . If only using the expression (1.4) of , 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 , which may be the geometry structure expected by Bombieri (2000), and proposed a basic framework of proving RH by concise geometric analysis.
2. Find Local Geometry Property by Computing
The norm is used in complex analysis. If is small, then is also small. To enlarge the role of v, in critical region we introduce a strong norm
in which and are of same order and is stable for . Note that if , , , then , but , because , see Figure 2.
It is known that has the exponential decay  
To make the figures of , we take a changing scale
and consider the curves and strong norm , later no longer explain.
Figure 1. The curves and , .
Figure 2. Figures of curves , , and .
“positive phase-difference”, so (i.e. RH is locally valid). But we do not know how to describe it, and it’s pity, is given up. At that time we always wanted to study no zero of the infinite series by the asymptotic analysis. Many times attempts have failed. After three years, we have suddenly waken up that.
“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 for . It is seen that in each root-interval , the imaginary part v has opposite signs at two end-points of and at some inner point , then is a valley curve. Therefore form local peak-valley structure and . Although each lower bound is different for different and , but which is always positive.
3. Local Peak-Valley Structure in Single Peak Case
For fixed the zeros of form an infinite sequence dependent on
We shall take them as the base and consider only single peak case.
The slope of single peak. For any , there are from negative peak to positive one, and from positive peak to negative one.
Theorem 1 (single peak case). Assume that is the single peak for any , then for any .
Proof. Below it is enough to discuss inside each root-interval . For any fixed , using Lemma 1, we consider two cases as follows.
As near the left node , we have
As near the right node , similarly
which are valid and numerically stable for .
Because has opposite signs at two end-points in , there certainly exists an inner point such that . Thus form a local peak-valley structure. We regard as a continuous function with respect to , which certainly has a positive lower bound independent of ,
This is a fine local geometric analysis.
Thus in each root-interval , we can determine a positive lower bound , which form the positive infinite sequence
Because each t must lie in some , thus for any t. In this way, we have completely avoided the summation process of the infinite series . Theorem 1 is proved.
In the theorem 1, we are anxious that when increases, if some root-interval will be contracted to a point so that ? This worry is denied by the following theorem 2.
Theorem 2. Assume that for , the root-interval is far less than two adjacent root-intervals, then when increases, the corresponding root-interval will enlarge, rather than decrease.
Proof. Assume that for , has a solitary small root-interval , , , and attains maximum value at some inner point , and at . We consider a little large sub-interval , in which , and is a convex curve upward. So for , and for .
Consider a small increment . By basic expression in I we have
Thus the real part has removed in parallel by a distance toward its convex direction. Due to outside , there are certainly a left node with and a right node with . Then is a positive peak curve inside the enlarged interval .
Besides, by basic expression we know that
i.e., still is a valley curve and in . Theorem 2 is proved.
From Theorem 2, we found a wonderful property:
When increases, these root-intervals have a tendency to get more unform.
This property makes RH be still valid when increases (e.g. RH is valid for ).
Finally summarizing the theorem 1 and theorem 2 our main theorem is proved.
Remark 1. We have a question: Is the single peak? Through large scale computation, Lune et al.   have pointed out that all zeros of u on critical line are single, no double. We can see in several curve figures that the is single peak, no multiple peak. This is heuristic, but not proof.
Remark 2. It is proved  that multiple peak case does not exist. For this, we
have used Lagarias’s positivity  to judge that
is not satisfied at multiple peak point. While Lagarias have used Hadamard’s formula independent of (1.4) to prove the equivalence. Can we directly get the judgment by (1.4)? No success, because we can get only linear relations by (1.4), but is a quadratic form. One unsolved question remains.
The author expresses sincere thanks to the reviewer for his valuable comments, suggestion and kind encouragement.
 Lune, J., Riele, H. and Winter, D. (1986) On the Zeros of the Riemann Zeta Function in the Critical Strip. Part 4. Mathematics of Computation, 46, 667-681.