Chapter 680 The Road to the Quasi-Riemann Hypothesis
Thoughts were flowing in his mind, and Xu Chuan froze there. A vague road appeared in his dilated pupils.
The Riemann Hypothesis is a question raised for the purpose of studying the π(x) function. It is a conjecture about the zero-point distribution of the Riemann ζ function ζ(s).
When Riemann was appointed as a corresponding member of the Berlin Academy of Sciences in 1859, as a meeting gift, Riemann submitted his only paper on number theory, and it was also the only paper that did not contain geometric concepts at all: "On the Individuality of Prime Numbers Less than a Given Value"
number".
This paper is not long, only nine pages, but it can be said that it has created a new era of analytic number theory in the history of mathematics.
In the paper, Riemann gave the accurate expression of the prime number counting function: π(x)=∞∑n=1·μ(n)/n·J(n?x).
There is no doubt that this is the core of the distribution results of prime number functions.
If the Riemann Hypothesis made him famous around the world, then by introducing the Riemann zeta function method, it was a truly pioneering work to elevate the study of π(x) from the real straight line to the complex plane.
Using the method of complex analysis, he combined algebra and geometry, pioneered the development of modern branches of mathematics such as topology and differential geometry, and brought the development of algebra into the field of the fourth dimension.
By using curvature to define the concept of space, Riemann created a new field of non-Euclidean geometry and was undoubtedly a true master of mathematics.
Of course, what made him famous around the world was the Riemann Hypothesis.
This conjecture of the century, defined as one of the seven millennium problems by the Clay Mathematics Institute, involves thousands of mathematical formulas based on it.
If the Riemann Hypothesis comes true, then at least more than two thousand mathematical formulas will be promoted to theorems; if the Riemann Hypothesis is proven false, it will subvert the entire mathematical world!
For Xu Chuan, what he was thinking about today was not this, but something he had studied when he went to St. Petersburg to attend the National Congress of Mathematicians last year.
The correlation function ‘random Hermitian matrix eigenvalue’ caused by the Riemann Hypothesis!
If the polynomial function on the yoke matrix is referenced through multi-complex variable function theory, Jensen polynomials and Taylor/McLaughlin series can be derived...
Maybe, he knows what to do!
.......
The thoughts and fragments in my mind are constantly being spliced together, and a shadowy road emerges in my eyes.
The radiating black pupils gradually condensed back, and Xu Chuan's eyes shone with joy. After his thoughts returned, he excitedly grabbed the arm of the figure in front of him, gave him a warm hug, and spoke incoherently in excitement.
"Hahahaha, I found it, I know it! I know what to do!"
The excited voice with a wanton smile echoed throughout the office.
On one side, Liu Jiaxin, who was hugged by Xu Chuan, stiffened for a moment. Feeling the heat and strength coming from her body, a blush spread quickly on her face, reaching the roots of her ears.
In the excitement, Xu Chuan didn't care about this. He quickly let go of the other party and said quickly: "Jiaxin, help me find a room and lend me manuscript paper!"
The inspiration in his mind had reached its peak at this moment, and he no longer cared about where he was.
Not only the Riemann Hypothesis, but also the Riemann Hypothesis and the pair correlation function of the eigenvalues of the random Hermitian matrix also made him unable to ignore it.
It corresponds to a function in physics that describes the energy level distribution of a multi-particle system under interaction. If there are no problems with his previous research, perhaps, in the field of number theory, he can come into contact with the fascinating love building.
Instein Rosen Bridge'!
.......
Late at night, in the building of Chuanhai Network Technology Co., Ltd., in the small cubicle next to Liu Jiaxin's office, under the bright light, Xu Chuan's pupils were bloodshot, but his face was full of excitement.
The tip of the pen touched the paper gently, holding the ballpoint pen in his hand, and quickly wrote out mathematical formulas and basic calculation theories one by one on the white A4 paper.
The thick stack of manuscript paper in front of me is covered with mathematical formulas, and the ground is littered with crumpled waste paper.
【π(x)=∫2x·dt/ln t O(x^1 2 e).】
This is the asymptotic formula of the π(x) function. Through it, the Riemann Hypothesis can also be further derived: [ζ(s)=np(1-p^(-s))^-1]
But now, what Xu Chuan wants to do is not to expand the Riemann Hypothesis through the asymptotic formula, but to further expand and compress it through the multi-complex variable function theory.
The Riemann Hypothesis is not that easy to solve. Before heading towards this mountain, which can be said to be the largest in mathematics, he still needs a tool to solve the problem of shrinking Re(s) to the number 1/2.
1/2, or 0.5, this number is quite special in the Riemann Hypothesis.
Since the Riemann Hypothesis was proposed in the 19th century, countless mathematicians have been fascinated by it.
During a long period of research, mathematicians called the straight line with Re(s)=1/2 on the complex plane the critical line.
Therefore, the Riemann Hypothesis can also be expressed as: all non-trivial zeros of the Riemann zeta function are located at the critical point of Re(s), and the real roots of non-trivial zeros are all 1/2.
Putting aside the mathematical rigor and logic, in the simplest terms, you can understand it as: "According to an important mathematical formula, we can draw many infinite points."
"Some of these points are arranged in a horizontal line, and other parts are arranged in a vertical line, but all the points are on these two lines, and none of them slips through the net."
The Riemann Hypothesis is such a mathematical formula, and one of the lines is a straight line based on 1/2.
However, since there are infinitely many of these points, there is theoretically no way to prove whether all points are on these two lines, because it can never be verified completely.
On the other hand, as long as a point is found that is not on the line, the Riemann hypothesis is overturned.
But so far, the mathematical community has used computers to verify that the first 1.5 billion such points all conform to the arrangement of the Riemann Hypothesis.
No one can find a point that is not online.
Therefore, usually, the Riemann Hypothesis is regarded as a theorem in the mathematical world, and many mathematical formulas are established based on its establishment.
.......
A long time passed by little by little without realizing it. The light in the cubicle was bright, and Xu Chuan didn't know what time it was.
[When Re(s)≤0,ζ(s)=2?π^8-1·sinπ8/2Г(1-s)ζ(1-s)】
Holding the ballpoint pen in his hand and quickly writing a mathematical formula on the paper, he fell into deep thought.
After a while, he scratched his head and paused the pen in his hand with some "annoyance" and "happiness".
After being reminded by his senior student Liu Jiaxin, he found the problem he had studied before, and also vaguely found some direction in his previous study of Einstein's Rosen Bridge.
But by mistake, he didn't find any ideas in the direction of his intended research. Instead, he got some inspiration from the Riemann Hypothesis.
Looking at the manuscript paper spread out on his desk, Xu Chuan pursed his lips. This is the derivation of the ζ(s) function and the ζ(1-s) function through the Poisson summation formula. It is the derivation of the Re(s) function.
When ≤0, it is just one of the core steps to prove trivial zero.
In layman's terms, it means to weaken the Riemann Hypothesis, and then solve the weakened Riemann Hypothesis, that is, the weak Riemann Hypothesis.
This is actually what the modern mathematics community has been doing.
Studying the lower bound number of the proportion of zero points on the critical line is the best method recognized by the mathematical community since the emergence of the Riemann Hypothesis critical zone idea.
In the ζ function of Riemann's hypothesis, all non-trivial zero points are located at the critical point of Re(s), and the real roots of non-trivial zero points are all 1/2.
This is conjecture and has not been proven yet.
But for now, the mathematical community has managed to summarize all the non-trivial zeros of the ζ function of the Riemann Hypothesis into the critical band of 0-1, which is close to 0.5.
To put it simply, I cannot yet prove that its real roots are all 1/2, so I will prove that they are all between 0-1.
Although this is not very standard, it is at least easier to understand.
The lower bound of the critical zone idea is such an idea.
By continuously advancing the distance of 0-0.5, the non-trivial zero points are gradually closer to 1/2.
On this road, a large number of achievements have emerged in the field of mathematics.
For example, in 1975, Levinson of MIT proved No(T)>0.3474N(T) before he died of cancer. In 1980, Chinese mathematicians Lou Shituo and Yao Qi studied Levinson's
The work improved a bit and they proved No(T)>0.35N(T).
At present, the best result of the research on Riemann Hypothesis is proved by the method of continuously approaching the critical zone.
But unfortunately, in the century and a half since the Riemann Hypothesis was proposed, research progress on the Riemann Hypothesis, including work on advancing the critical zone, is still far away.
Xu Chuan doesn't know if this is the right path, but for now, he seems to have found another way to get close to the non-trivial zero point.
Although this is just a little bit of thinking and needs to be continuously improved in the future, it can be said that if this idea is released by him, it will definitely shock the entire mathematical world and set off a wave of Riemann Hypothesis.
However, this is not what he wants.
The 'random Hermitian matrix eigenvalue' pair correlation function he wanted to study has not made much progress today.
He even had an intuition somewhere. Perhaps only by completely solving the problem of Riemann's hypothesis could he have access to the secret of 'space-time'?
Prime numbers may really be connected to space and time, hiding the deepest mysteries of the universe.
...
PS: I just started working in the new year and I am a bit busy. As expected, I have to work overtime. In addition, I have recently read the papers on the Riemann Hypothesis and Space-time Wormholes and I feel bald. I am stuck thinking about it. This is to make up for yesterday.
