Local Geometric Proof of Riemann Conjecture

Riemann hypothesis (RH) is a difficult problem. So far one doesn’t know how to go about it. Studying ζ and using analysis method likely are two incor-rect guides. Actually, a unique hope may study Riemann function by geometric analysis, which has the symmetry: We show that u is single peak in each root-interval of u for fixed . Using the slope t u , we prove that v has opposite signs at two end-points of j I . There surely exists an inner point such that 0 v = , so { } , u v β form a local peak-valley structure, and have positive lower bound ( , j I . Because each t must lie in some j I , then 0 ξ > is valid for any t (i.e. RH is true). Using the positivity ξ ξ ′ > of Lagarias (1999), we show the strict monotone , and the peak-valley structure is equiva-lent to RH, which may be the geometric model expected by Bombieri (2000). This research follows Liuhui’s methodology: “Computing can detect the unknown and method”.

and transforming the integral by which is already analytically extended to the whole complex plane except for 0,1 s = . Clearly, the pole points 2, 4, 6, s = − − −  of ( ) RH is a very difficult problem, which has stimulated the untiring research in the areas of the analytic number theory and the complex functions, even the scientific computation. Smale [1] (1998) reported 18 mathematical problems for next century. The first one is RH. Cray Mathematics institute (2000) announced the seven problems of the Millennium, in which RH is reviewed by Bombieri [2].
There have been many theoretical researches for RH, e.g. the reviews [2] [3] and books [4] [5]. A lot of numerical experiments verified that RH is valid.
However, RH has not been proved to be true or false in theory.

Theoretical Research
We list some important progressions as follows, see [4].
3) The moment method of ζ . Levinson (1975) proved that a number of the roots on critical line attains 34.74%. Conrey (1989) improved to 40%, and then he [3] (2003) pointed out that 99% of all roots lies in 8 n 1 2 l t σ − ≤ . This is the best result up to now.
We have seen that except for critical line, most works focus to ζ . We think that the appointing ζ in formulation of RH is a historic misguide, because which has gone against the original thinking of Riemann (Actually, he focused to ζ , rather than ζ , see 5, and ζ behaves badly, see 6). This is likely the first misguide in studying RH.
In recent twenty years, many new methods appeared and the research of RH has taken some important progressions. But RH has not been solved yet. 1) There are a high peak in each segment of the graph and 1 -9 smaller peaks between two high peaks. They found that the ratio of the high-peak and low-peak can reach 1000 times.
2) There are 1 -8 roots between two high-peaks. They found a pair of large zeros, these two zeros are very close to each other, and look like a double zero. e.g. in Fig.7, p.678, [8], the spacing between two zeros is less than 0.00011, and the value ( ) To face so terrible micro-structures near critical line, we has always met a wide gap: how to prove no zeros of the infinite series, analysis method is powerless. Corney [3] (2003) pointed out that "It is my belief, RH is a genuinely arithmetic question that likely will not succumb to methods of analysis". This likely is the second misguide in studying RH. We should give up the infinite summation analysis.

A unique Hope is to Study ξ
Although ζ diverges for ( ) 1 Re s ≤ , but can be estimated as follows, see [5] ( son, analysis method is also powerless for ξ . Bombieri [2] (2000) pointed out that "We do not have algebraic and geometric models to guide our thinking, and entirely new ideas may be needed to study these intriguing objects". This is a valuable inspiration. We hope to establish a geometric framework.
We know that ξ has the most important symmetry on critical line. We point out that the positivity Re ξ ξ ′ > of Lagarias [9] (1999) is an essential progression for ξ , which also is a unique result to be cited in our proof for RH, see 3. Assume that RH is true, denoting u iv ξ = + and 0, for 0, plays an important role in our proof.

Local Geometric Model for ξ
From these difficulties and advices of Conrey and Bombieri, we should give up ζ -function and pure analysis methods, while turn to geometric analysis. What is geometric analysis? We no longer regard the summation process of series, while prefer the geometric property and structure of ξ -curve itself. That is, "Explain the essence by figure" (Liuhui's words). This is a big change of our recognition. We compute and study by Liuhui thinking, i.e. "computing can detect the unknown and method" (see 7), finally find a local geometric model for u iv ξ = + , which contains four basic concepts as follows.  , 0 u t β > inside j I . 2. Single peak. If u in each root-interval j I only has one peak, called single peak, else called multiple peak (It is proved that the multiple peak case does not exist, see theorem 2).
The single peak u has the following geometry property.
3. Slope t u . For single peak u and any 0 β ≥ , there are 0 t u > from negative peak to positive one, and 0 t u < from positive peak to negative one.
Sequence principle. As the zeros of u do not have finite condensation point, each t must lie in some j I , then 0 ξ > is valid for any t.
We have gotten Basic Theorem. All zeros of Riemann ξ -function lie on the critical line.
Besides, by (1.12) we have Equivalence theorem. The peak-valley structure and RH are equivalent.
We think that the strict monotone is the deepest description for RH. The PVS may be the geometric model to be expected by Bombieri, which makes the proof of RH get concise and intuitive, and many difficulties are avoided, e.g. need not discuss the summation process of the infinite series and so on. The PVS and RH in single peak case were shown in our previous paper [10].
This paper wants to give a full proof of RH, including PVS, nonexistence of multiple peak case, the equivalence and strict monotone. I think that I have realized the original thinking line of Riemann, see 5. Beside we also add the reasons to give up ζ in 6. Why I want to study RH? My initial aim is to examine that can Liuhui's thinking solve the most difficult problem? which makes me persist in studying in whole four years.

Detect Local Peak-Valley Structure by Computing
here and below the Cauchy-Riemann conditions are used many times. If 0 β = , we have the following analytic property.
The symmetry. If 0 β = , then where three conditions of norm are satisfied. Its advantage is that u and v β are of the same order and ξ is stable with respect to Figure 2.
Firstly we compute u iv ξ = + . Take a changing scale ( ) , when drawing curves of ξ , x-axis is t, y-axis is u M such that 1 u M ≤ . No longer explain later. Figure   1 exhibits the curve ( )  , Nextly we compute the derivative We see that t u and t v have also alternative zeros and a local PVS in Figure 3.

Local Geometric Proof of RH
We regard β as a continuous changing process from 0 β = to which have not the finite condensation points, else 0 u ≡ . We take them as the base in studying PVS.
Theorem 1 (single peak case). If ( ) , u t β is single peak for any ( ]  . For any fixed 0 β > , using the analytic property (1.12), we consider two cases as follows.
They are valid and numerically stable for ( ] This is a fine local geometric analysis.
Thus in each root-interval Because each t must lie in some j I , thus 0 ξ > for any t. In this way, the summation process of the infinite series ξ is completely avoided.  Theorem 2. The multiple peak case does not exist for [ ] Figure 4. Clearly, 0 u > in Below we prove that the multiple peak case does not appear. For this, we consider the minimum extreme value ( ) On the other hand, because RH is valid as before, using (1.12), we have , u t β for any 0 β > is single peak and its limit ( ) ,0 u t is also single peak.  Why need to deny the multiple peak case? We see in Figure 4 that when β grows, the curve ( ) , 0 u t β > near 2 j t t = will decrease towards its local convex direction (see proof in theorem 3). This will bring a dangerous possibility to be close to t-axis such that Theorem 3. The peak curve ( ) ,0 u t in a small root-interval (including double root) will remove in parallel towards its convex direction for 0 β > so that ( ) Proof. Assume that ( )    Besides, by basic expression and the slope t u , we know So there surely exists an inner point * Lune et al. [8] pointed out that all roots on critical line are single, no double.
Maybe, in the future, some double roots are found, but in this case, theorem 3 still confirms 0 ξ > for 0 β > . This is one of the most mysterious property for ξ .
Summarizing three theorems above, our basic theorem is proved.  Remark. In the proof of Theorem 1 we have seen that the Riemann integral ( ) K f ξ = has the symmetry, which is independent of the speciality of f. So  Haglund [11] has discussed Ξ and other functions with numerical experiments, and proposed a conjecture: If function N F has monotonic zeros, then which implies RH. Sarnak [12] has analyzed the Grand RH of L-function, which are more difficult.

Lagarias Theorem and Other Conclusions
In the proof of RH, we have used a unique new result to be the following.
Lagarias theorem (1999). If RH is true, then This is a unique equivalence to RH for ξ , we think that this is an essential progression along research line of ξ after Hadamard (1893) and Mongoldt Taking logarithm and derivation, we have  Both are equivalent. However, the local geometry property is of extreme importance, which makes the proof be concise and intuitive. I greatly appreciate the mathematical beauty of the symmetry.

Follow Riemann Thinking
Riemann's paper "On the number of primes less than a given magnitude" is a classic work [5], we consider a part of it, and give remarks with 5 items.

1) In fact
This function is finite for all finite values of t and can be developed as a power series in tt which converges very rapidly. t π as t grows, this expression converges and for infinite t is only infinite like log t t; Thus it differs from ( ) log t ξ by a function of tt which is continuous and finite for finite t and which, when divided by tt, is infinitely small for infinite t. This difference is therefore a constant, the value of which can be determined by set- Remark. 1) It is strange that Riemann had not given a complete integral expression (1.6) of ( ) s ξ . I think this is a neglect, which will bring misunders- 1) In these 5 items, Riemann had always discussed ξ , rather than ζ . I don't know that from what time, RH had become to study ζ . This has gone against the original thinking of Riemann and was a historic misguide. Maybe, there is a possibility that due to the decay of ξ , computing ξ is too hard, while computing ζ is easier, but analysis method for ζ is hopeless. Actually, a unique hope is to study ξ . Author (2020) found PVS and proved RH by local geometric analysis for ξ .
What geometry structure it is? No PVS.
To draw curves, we use the norm ( ) s U V ζ = + . Edwards [5]    0.005, 0.01 β = . This is a mystery to be hidden behind ζ , which consists with theorem 3.

Guide Role of Liuhui Thinking in Proving RH
Greek preferred the deduction. To face so difficult RH, where did the idea of proof come from? The Greek mathematics does not give any inspiration. But the eastern mathematics may do it! Because Chinese emphasized the combination of computing and analyzing, in particular, Liuhui thought that "can detect the unknown and method by computing". We shall show how to reveal the essence of RH by Liuhui methodology.

Liuhui Was the Greatest Mathematician in Ancient China
Chinese "Nine Chapters Mathematics" (at least B.C.4-2 century) and Greek  [13], and basically formed a mathematical system including geometry, computation, algebra and analysis. By analyzing these remarks, we found that his deep idea had already surpassed that period so that cannot be understood by posterity, and forgotten about 1500 years. Up to recent 50 years, it is gradually recognized newly [14]. We list five items of Liuhui's discovery as follows.
1) Sum and take limit to prove the existence of π.
In B.C.11 century, Shanggao theorem (Pythagoras theorem, in B.C.5 century) was proposed and extensively applied. Liuhui in "cut circle" (1600 words. English version [15]) had computed the area of 96-polygons to get 96 3.14 π = (the area is more intuitive than the circumference) and then 3072 3.1416 π = . He for first time had proposed the limit concept: "The more finely is cut, the less loss there is. Cut it again and again until one is unable to cut further, that is, when the shape coincides with that of the circle and there is no loss".
Liuhui considered the ratio of two small squares (found by Wang [17], 1996)  the same as the previous one; in this way, the ratio is again verified". This is the prediction-correction idea.
The extrapolation is an important idea of modern scientific computation. We [18] proposed the extrapolation prediction multiple grid method to solve PDE. 3 V R = π by this method. The both play an important role in completing Chinese mathematical system (although Archimedes has already obtained). While Liu-Zu principle, "the area and hight are same, then their volume is also same", called Cavalieli principle (1635) in the west. 5) Use "rate" and "multiple difference" to study the ratio of difference, the seed of the "slope". Ancient Greek did not have these concepts.
"Nine Chapters" discussed practical problems, the "relation" of two quantities just is function, and the piecewise expression of function was used. While the "rate" (called "lv" in ancient China) is difference ratio of function (averaging slope or velocity) has the rate A.
The linear interpolation was extensively applied in "Nine Chapters". Fibonacci We have seen that Chinese ancient mathematical thinking was very different from Greek. In that period, Liuhui had studied these two "infinitesimal processes". But nobody could understand his thinking 50 years ago.

Liuhui Methodology (Historic Contribution)
Liuhui's preface (800 words) is a wonderful paper through ages, and expounded his deep mathematical thinking.
1) "Scientific discovery is a recognition process of the prediction and correction". Liuhui's extrapolation is a typical prediction method.
2) Mathematical methodology: "Computing can distinguish tiny and detect the unknown and method". "And analyze the reason by logic, explain the essence by figures".
This is an open, progressive and creative methodology (do not stipulate conclusion in advance), very different from Greek idea. Chinese academician Z. C.
Shi pointed out [19]: "Scientific computation, experiments and theory, like a tripod, supplement each other, and become three methodologies in modern science actions", and emphasized: "Scientific computation not only is a numerical method, but also a method of research". Liuhui's idea just is third method, which plays an important guide role in studying modern mathematics [14].
When study a difficult problem, because for lack of understanding, we should detect its properties by computing in various directions, then analyze these results and look for key properties, finally find the idea of proof. This just is Liuhui thinking.

Detect the Essence of RH by Liuhui Thinking
I studied ancient Chinese mathematical thinking for 10 years. I believe that Liuhui methodology has powerful vitality. How to detect the mystery of RH? During four years, my recognition process is as follows.
1) Studying ζ is hopeless, see 6. A unique hope is to study ξ. 2) We found that the real and image parts of ξ have the positive phase-difference, which implies RH. But we don't know how to describe it. A chance of local geometry analysis is missed. At that time we always wanted to use the asymptotic analysis.
3) To attain the exponential decay ( ) 4 e t O − π , we used Riemann technique of integration by parts and Jacobi equality to get high-order expression, , using a comparison criterion, it seems RH is proved. But for large t, the asymptotic analysis is also hopeless.

5)
In these researches, we have always met a wide gap: How to prove no zero for the infinite series? It is impossible. Finally, in 2019 Oct.2, we have suddenly waked up that Give up the summation process, study the geometry structure of ξ itself. Actually we come back to the positive phase-difference once again. But now, we have found the local peak-valley structure and the importance of symmetry, and then proved RH by geometry method, where t is arbitrary.