Great Country Academician

In the villa at the foot of Purple Mountain, Xu Chuan was obsessed with studying the Riemann Hypothesis.

Although it is said that he has found a way to the weak Riemann Hypothesis, it is still unknown whether he can ultimately solve this problem.

Moreover, even if this idea is effective and can continue to advance the critical zone of the Riemann Hypothesis, it is not easy to continue to narrow and solve it.

Mathematicians often write the real part and imaginary part of the non-trivial zero point of the Riemann zeta function as σ and t respectively. The vertical strip with 0 \u003cσ \u003c 1 on the complex plane is called the critical zone, and the vertical strip with σ = 1/2 is called the critical zone. The line is called the critical line.

As early as when Bornhard Riemann wrote the paper "On the Number of Prime Numbers Less than a Given Value", it was given that all non-trivial zeros of the Riemann zeta function are located on the critical line of 1/2 superior.

When subsequent mathematicians conducted targeted research, because it was too difficult to prove that non-trivial zero points are located at the critical line of 1/2, they extended it to 0＜Re(s)＞1, hoping to prove that all non-trivial zero points are located at 1/2. lies on this critical zone.

Regarding this point, what is interesting is that Riemann had already given an accurate answer in his original paper.

As for the reason, maybe it’s because of disdain? Think this is too easy to be on a paper?

Or maybe it's like the famous saying written by Pierre de Fermat, the French mathematician who proposed Fermat's conjecture in the seventeenth century, when he read the Latin translation of Diophantus' Arithmetic.

"Regarding this (this refers to the later Fermat's Last Theorem), I am convinced that I have discovered a wonderful proof. Unfortunately, the blank space here is too small to write it down."

In the paper "On the Number of Prime Numbers Less than a Given Value" written by Riemann, there are many similar words.

Many important places where the detailed process should have been written were eventually replaced by the sentence "proof omitted".

Otherwise, the paper he presented to the Berlin Academy of Sciences would not have been only eight pages.

Of course, it can be said that almost all scholars have done this by using similar words like ‘omitting proof’ to save the length of a paper.

Including Xu Chuan himself, he also used numerous simplified calculation steps in the papers he proved.

But whether it is him or other mathematicians, the method of "omitting proof" is generally used to omit obvious proofs.

But Riemann was different. This was not the case in his paper. Some of the "omitted proofs" he wrote in that eight-page paper took decades of efforts by later mathematicians to complete, and some even It remains blank to this day.

Just like it still took scholars of later generations several decades to completely rule out the absence of non-trivial zero points in the two regions of the Riemann function Re(s)=0 and Re(s)=1.

Including the advancement of critical zones, they are also proposed and studied based on this.

If someone asks, apart from proving the Riemann Hypothesis, does compressing the critical band and bringing the non-trivial zero point close to 1/2 have any other benefits?

The mathematical community will tell you that the prime number theorem of later generations was proved based on the fact that there are no non-trivial zero points in the two regions of the Riemann function Re(s)=0 and Re(s)=1.

As for the importance of the prime number theorem, it goes without saying.

Today's network passwords involving computer security are largely based on the prime number theorem.

In addition, many aspects such as industry and agriculture are also inseparable from prime numbers.

For example, in many high-precision gear designs, the size of the transmission gears, one large and one small, is closely related to prime numbers. Simply put, prime number design can increase the durability of gears and reduce mechanical failures.

Of course, for many mathematicians, they study mathematics not because of its application capabilities. Rather it is there.

Including Xu Chuan, if the Riemann Hypothesis he is currently studying is truly confirmed, will it bring about earth-shaking changes to the entire world?

Not really.

On the one hand, the Riemann Hypothesis has always been used as a theorem by the mathematical community.

On the other hand, even if the Riemann Hypothesis involves many fields such as cryptography, it will take an extremely long time to turn the theoretical results into applications and develop various related uses.

This time is calculated in units of ten years or even longer.

For example, the Poincaré conjecture, Hodge conjecture, NS equation, Yang-Mills existence and mass gap, which are also among the seven millennium problems, have been solved for a long time.

Especially the Poincaré conjecture, it has been more than 20 years since it was proved by Perreman in 2003. But it can only be used in computers, medical treatment, industry, etc.

As for the next three solved by Xu Chuan, apart from establishing a control model for ultra-high temperature and high pressure plasma turbulence based on the NS equation, there are still few applications in other fields.

Mathematics is such a pure science.

Many times, mathematicians study mathematics not for the many applications it can have, but for the worldly truths hidden in the wonderful mathematical formulas!

In the study, Xu Chuan turned on the light and placed a paper on the Riemann Hypothesis that he had printed out not long ago in the corner.

Over there, you can see nearly half a meter of paper piled up, which he has read over the past few days.

Of course, he had not read all the papers in detail. Some of them had only been briefly browsed, looking for something valuable.

These days, in order to help him understand the Riemann Hypothesis more deeply and solve this century's problem, he has collected a large number of papers on this aspect.

Not only papers related to the Riemann zeta function and non-trivial zero points, but also papers related to the π(x) function and the 'random Hermitian matrix eigenvalue' pair correlation function.

He even made a special call to his mentor Pierre Deligne.

When he heard on the phone what Xu Chuan was currently studying, the expression on the face of this old man who usually didn't care much about anything except mathematics suddenly changed, and his breathing became rapid.

After coming back from his daze, Deligne ignored the shock in his heart and quickly asked: "Are you studying the Riemann Hypothesis?"

"Um."

Xu Chuan nodded and responded. The only people who could communicate with him about this kind of mathematical research were the people at the top of the pyramid.

Although his mentor, Pierre Deligne, inherited from Mr. Pope Grothendieck, his main research field was algebraic geometry, he also had great strength in number theory.

For example, the Weil Hypothesis proved by his old man is the Riemann Hypothesis on the elliptic curve.

Although this problem was planned in the field of algebraic geometry, it is undoubtedly one of the most brilliant achievements in the field of pure mathematics. His knowledge in the field of algebra is naturally extremely strong.

Of course, if we want to say that the strongest person in algebraic geometry and number theory today, aside from himself, he should be Professor G. Faltings.

Even in the field of number theory, Xu Chuan felt that he was not necessarily as powerful as Professor Faltings.

After all, this man is a master who directly used algebraic geometry methods to prove Model's conjecture in number theory, as well as complete the Riemann-Rohe theorem of arithmetic surfaces and other algebraic problems.

Since he rose to prominence in academia, Faltings was the only one who allowed him to revise his thesis.

In the previous proof paper of Verberg's conjecture, this arrogant scholar from Germany proposed many things that needed to be revised.

However, in terms of relationship, his relationship with Professor Faltings is definitely not as good as that of his mentor Deligne.

Therefore, at the first opportunity, the person to discuss related issues naturally fell on Professor Deligne.

As for another mentor, Edward Witten, although he won the Fields Medal, he was not a scholar in the field of pure mathematics, and he did very little research on pure mathematics.

On the other side, on a wicker chair in the park of the Institute for Advanced Study, Deligne, who was originally walking in the park, was no longer in a relaxed mood.

After confirming that his student was really studying the Riemann Hypothesis, he quickly asked: "Do you have an idea? How far has it progressed?"

He is very clear about his student's character. Judging from the past, once his student officially starts researching a certain mathematical problem, it can be said that he has basically a certain grasp, or idea.

Even, to a certain extent, when he begins to formally study a certain mathematical problem, the solution may not be too far away.

Although this sounds incredible, after all, at their level, the research problems are almost all the world's top conjectures or difficult problems. No one dares to say that they will be able to produce results, but this student is a "counterexample freak" '.

It can be said that the problems he targeted were finally solved.

Hodge conjecture, NS equation, Yang-Mills existence and mass gap problem

If the Riemann Hypothesis is solved again, he will have solved four of the seven most famous millennium problems of the 21st century by himself.

Looking at Deligne who was eager to know the answer during the video call, Xu Chuan smiled and said: "I have a little idea. At present, it is still far from the Riemann Hypothesis, but it is possible to continue to compress the critical zone. It's possible."

"Compression critical zone?"

Hearing this, Deligne thought for a moment and replied: "The current research on the critical zone has been completely proven that No (T) \u003e 0.35N. The controversial one was previously proved by Professor Walter Jeffrey of Harvard University. No(T)\u003e0.4N, what step have you taken?”

Although the Riemann Hypothesis is not in his current research scope, as a scholar who has solved the Weil Hypothesis (Riemann Hypothesis on elliptic curves), he is naturally aware of the current progress of the Riemann Hypothesis in mathematics.

The idea of ​​​​squeezing the critical band is the most commonly used and most effective proof method in today's mathematics world. Xu Chuan used this method to study the Riemann Hypothesis, which was not surprising to him.

On the opposite side, Xu Chuan shook his head and said: "The idea of ​​continuing to compress the critical zone is indeed feasible, but I am not prepared to do so."

Hearing this, Deligne suddenly showed a surprised look on his face: "How do you say that?"

After thinking for a moment, Xu Chuan said, "Intuition, right?"

After a slight pause, he continued: "In recent days, I have read a lot of research and papers on the Riemann Hypothesis, and many of the results are based on the idea of ​​compressing the critical band."

"It is undeniable that these results are indeed outstanding. But in my personal opinion, it is too difficult to compress the Riemann zeta function and non-trivial zero points to the number 1/2. Or, it can even be done Said there was no hope."

"After all, prime numbers are infinite, and the number of non-trivial zero points is also infinite. This alone is enough to block the current research ideas of compressed critical bands."

“Maybe we can continue to advance on this path, and even push it to 0.45, 0.46 or even higher. But if we want to stably compress it to 1, I don’t think there is much hope.”

“At least there’s little hope with traditional research methods at the moment.”

For Xu Chuan, reading the papers these days is not in vain.

Although there wasn't much that was helpful, he knew quite clearly about methods to compress the critical band and increase the number of non-trivial zeros on the critical band.

Intuition told him that although this method was very effective in studying the Riemann Hypothesis, if he wanted to rely on it to solve the Riemann Hypothesis and push the real root of the non-trivial zero point to 1/2, the feasibility was almost zero.

Otherwise, he does not need to find another way to find another method, he can just continue the previous research.

Listening to Xu Chuan's explanation, Deligne frowned, with some contemplation on his face.

By compressing the critical band and increasing the number and proportion of non-trivial zeros on the critical band, this method is one of the mainstream methods currently used in the mathematical community to study the Riemann Hypothesis, and it can even be said to be the mainstream method.

After the 21st century, more than two-thirds of the research on the Riemann Hypothesis is based on this method.

But even counting the controversial No(T)\u003e0.4N from Harvard University, they are actually far away from the ultimate goal of No(T)=N(T) (that is, all non-trivial zero points are on the critical line), and There is still a long way to go.

0.4-N(T), or 0.4-1, which is still a difference of 0.6.

Over the past century and a half, their progress can even be described as insignificant for the Riemann Hypothesis.

But in any case, compressing the critical band and increasing the number and proportion of non-trivial zero points in the critical band is still the best way to study the Riemann Hypothesis.

However, Xu Chuan now says that he is not prepared to use the traditional method of compressing the critical zone to study the Riemann Hypothesis, and even speculates that this research route may not work.

Although standing at his height, he rarely shakes his heart because of one or two unproven opinions, but this time he was indeed surprised by his own student.

Taking a deep breath, Deligne quickly said: "If it's convenient, can you tell me your research ideas?"

In academia, asking a scholar who is studying a difficult problem for research ideas is a taboo thing, even if this person is his student.

But at this moment, Deligne didn't care about these things.

After all, this is the Riemann Hypothesis, which is related to thousands of mathematical theorems!