Great Country Academician

Hearing Deligne's impatient inquiry, Xu Chuan was not surprised either.

He nodded slightly and said with a smile, "Of course."

Although in academia, it is an unspoken rule that most people will abide by not asking other scholars about the ideas of unfinished research, but it is obvious that he and Deligne do not fall into this category.

Neither he nor Deligne would do such a dirty thing.

After a slight pause, Xu Chuan organized his words thoughtfully and said: "If you want to advance the non-trivial zero points of the Riemann Hypothesis to 1/2, what you need to do is to prove that all the zero points are distributed on the zero boundary. But this is It is a difficult route to reach the end, and the infinite number of prime numbers separates the two worlds like a barrier in the universe."

"So, it seems to me that instead of working in the non-trivial zero region, it's better to shrink it back to Jensen's inequality and then work through subsigma algebra"

"Perhaps this path is more promising than shrinking the critical zone."

"Jensen's inequality" Deligne thought for a moment and asked quickly: "Your idea is to regress the prime number counting function π(x)?"

Although the Riemann Hypothesis was not the direction of his research, and it was difficult to think of any new breakthroughs, as a top mathematician, after Xu Chuan proposed his own research ideas, he was still able to see through the essence of the problem.

Xu Chuan nodded and said with a smile: "Yes, the zero points of Reimann ζ are inseparably related to prime numbers. The most direct one is that the prime number counting function π(x) can be represented by the zero points of ζ. And the prime number counting function is given The number of prime numbers less than or equal to x, such as π(10) = 4, because there are 4 prime numbers less than or equal to 10: 2, 3, 5, 7"

".Weierstrass's decomposition theorem can be seen as an extension of the fundamental theorem of algebra: that any whole function can be expressed as the product of functions related to its zeros."

During the video call, Xu Chuan and his mentor chatted about research ideas on the Riemann Hypothesis.

Across the way, in the park of the Institute for Advanced Study in Princeton, Deligne frowned and kept thinking.

After a while, he raised his head, stared at Xu Chuan with shining eyes, and said: "It is indeed a very interesting road, but how do you solve the problem that the inverse integral transformation cannot converge well at the jump of the π(x) function? "

Although analytic number theory is not the main area of ​​his research, he still understands the way to solve Weil's conjecture.

Perhaps it is not as good as Professor G. Faltings and Professor Jean-Pierre Serre who specialize in the field of number theory, but starting from Xu Chuan’s analysis, we can find some big problems that may exist behind this road. , it is not difficult for him.

Across the video, Xu Chuan smiled and shrugged, saying: "There is no good way yet."

As Deligne said, there are still many problems on his way to study the Riemann Hypothesis or solve the quasi-Riemann Hypothesis. For example, the inverse integral transformation cannot converge well at the jump of the π(x) function. These are problems that need to be solved. These are troubles that need to be solved.

But this is the Riemann Hypothesis. It is normal to have troubles and problems.

If it were easy to solve, it wouldn't have been around for a century and a half and would have been killed long ago.

Deligne thought for a moment and said, "Perhaps this may be more difficult than compressing the critical zone."

Different from the traditional method of studying the Riemann Hypothesis in today's mathematics community, the research idea of ​​regressing the prime counting function π(x) proposed by Xu Chuan is obviously a new method, or a relatively niche path, at least today Not many people in the mathematical community use this method to study the Riemann Hypothesis.

In this case, there is no way of knowing how many problems may be encountered before taking this path, and the difficulty may be greater than the method of compressing the critical zone.

Hearing Deligne's words, Xu Chuan smiled and said, "Whether it's difficult or not, it's all there. And"

"If the Riemann Hypothesis was not difficult, I would have no interest in research."

This sounds a bit crazy, but as he said, it is difficult to arouse his interest.

If it was solved casually, there would be no research significance for him.

Deligne nodded without asking anything, and instead said, "If you solve the Riemann Hypothesis, please remember to tell me as soon as possible."

He actually knows very well that for people like them, once they determine a certain direction, unless they really find something, they will never give up until they see the Yellow River, and they will never look back until they hit the south wall.

Even if they meet the Yellow River and hit the south wall, they will stubbornly choose to go in their own direction.

Just like himself, didn't he also pursue the basic objects of arithmetic, algebraic geometry, and study the standard conjectures left by Grothendieck for a lifetime.

Xu Chuan nodded: "If it can be solved, it will definitely be done."

After chatting with Deligne for a while, he asked him for some textbooks on flat cohomology and L-radical cohomology, non-Abelian algebraic geometry, duality between continuous and discrete, etc. left behind by his ancestor, Mr. Grothendieck. After writing the paper, Xu Chuan ended the video call.

Although he studied under Professor Pierre Deligne and was a disciple of the Pope, he had not read all the papers left by Mr. Grothendieck.

Because there are just too many.

He created a complete set of modern abstract theoretical systems of algebraic geometry. Not to mention others, there are more than a dozen masterpieces in the field of algebraic geometry alone.

In addition to the field of algebraic geometry, he is also proficient in topology, abstract algebraic geometry, general shapes, number theory and other fields.

Although in the later years of his academic career, he stayed away from the academic world due to his anti-war ties and prohibited the dissemination of all his works to the outside world, he still left behind a large number of papers and manuscripts.

Maybe others don't have it, but as one of Mr. Grothendieck's most outstanding disciples, Deligne must have a large number of papers and manuscripts in his hands.

And it was impossible for him to read all of these things when he was studying abroad at Princeton.

In addition, he was studying the Hodge conjecture at the time, and did not have much time to study Grothendieck's works.

But now, in order to solve the Riemann Hypothesis, Xu Chuan can only ask his mentor for one more time.

While Xu Chuan was flipping through the information to study the Riemann Hypothesis.

On the other side, on the Arxiv forum, there is an article about P=NP? The conjecture paper quietly set off a storm of discussion in the mathematics community.

The paper is exactly what my senior sister Liu Jiaxin passed on, "Proof of solving the factorization of positive integers with polynomial algorithm!" 》

Over the past few days, after perfecting the suggestions and questions Xu Chuan asked her, she uploaded the paper to arxiv.

Speaking of this, today's mathematics community is probably influenced by a guy who has gone into retreat. Many scholars already like to upload their papers to the preprint website Arxiv when submitting papers.

Many scholars, including many who didn't like this book that they considered to be full of "dregs", have now registered accounts and even occasionally read other people's papers and ideas.

But at the same time, arxiv has also attracted more "civilian" scholars, and one "outrageous" paper has been posted on it, hoping to cause some splash and repercussions, and gain some traffic.

In short, mathematics has gradually overtaken the computer papers that were originally popular on the arxiv website in the past two years, and once again occupied the top spot on the preprint website.

Such a paper related to the seven millennium problems naturally quickly attracted the attention of others.

On the famous international mathematics forum MathoverFlow

[Have any of you read the paper "Proof of Solving the Factorization of Large Positive Integers with Polynomial Algorithms" on arxiv? 】

【Great positive integer factorization problem? P=NP? guess? That Professor Xu? Did he take action again? 】

[No, the first author of the paper is called Jia Xin Liu, and he seems to be a scholar from China. 】

[Liu Jiaxin? I’ve never heard of this name (shrug). If it weren’t for Professor Xu, there are papers on this world-class problem on arxiv, so forget it. Are there not enough parallel imports and civil sciences of all kinds? 】

[Indeed, arxiv has become more and more mixed in the past two years. Although many mathematics experts have registered accounts, there are still many guys who know nothing. Papers on Goldbach's conjecture and papers on the grand unification of mathematics are everywhere now, and there are even papers proving 1+1. 】

[No, no, no, this paper is very different. My supervisor has already read it. He is currently studying it and says that it is very likely that it has really solved the factorization problem of large positive integers. 】

[Not everyone is qualified to say this. May I ask who your mentor is? let me see. 】

【Tao Zhexuan! 】

【.】

Scholars on the international mathematics forum MathoverFlow have paid attention to things, and naturally the top mathematicians have gradually noticed it.

University of California, Los Angeles, California, USA.

Tao Zhexuan was sitting in his office, flipping through the papers in his hands, his eyes hidden by black-rimmed glasses shining with sparkling energy.

As a young surfer in the mathematics world, he is probably the first among top mathematicians, not to mention the first, but he is definitely the first to discover this paper.

Of course, this also has something to do with his knowing Liu Jiaxin,

At the International Congress of Mathematicians in 2018, he met that girl. He was walking with that person at the time. He heard that she went to the same university as him. Later, he pursued her to study at Princeton, where she made many achievements. Achievements related to computer mathematics.

Although his reputation in the mathematics world is not very prominent, in the computer world, he is actually quite famous.

"It's a bit interesting. This is a twist on the quadratic sieve factorization method, and introduces the Hamiltonian graph judgment method and the polynomial function algorithm to make the judgment."

"This idea might actually be feasible."

"It's just that the shadow of this paper seems a bit familiar?"

Staring at the paper in his hand, Terence Tao muttered to himself.

Although it was impossible for him to understand all this top-level mathematics paper after reading it in one go, he had already flipped through it from the time it was printed out, and the paper in his hand had rough edges.

What concerned him was not just the solution to the problem of factorization of positive integers, but also the shadow of some ideas in this paper that looked very much like a certain person.

If it weren't for the fact that the name of the first author was indeed not that person, and if only some of the mathematical ideas were similar, he would even think that the paper in his hand was written by that person.

After flipping through the paper in his hand again, Terence Tao closed his eyes and thought for a while, then picked up the mobile phone on the table and opened the address book.

At the same time, on the other side, in China, in the study room of the villa, Xu Chuan was perfecting ideas for solving the weak Riemann hypothesis.

At this moment, his cell phone vibrated on the desk.

Looking up, he glanced at the caller information and answered the call with some surprise.

"Hey, why does Professor Tao have time to call me?"

With a smile and a joke, Xu Chuan asked. The person who called him was Terence Tao who was far away in the United States.

"Are you studying P=NP? Guess?"

Opposite me, Tao Zexuan didn't pay attention to Xu Chuan's teasing and asked straight to the point.

Hearing this, Xu Chuan was stunned for a moment, and then realized that it was probably the senior sister who had released the paper in her hand.

Although his name was not signed on the paper, although Liu Jiaxin wanted to put his name on the second work, after all, he provided a lot of help from the initial research to the publication of the paper, and even part of the ideas were his supply.

However, Xu Chuan still refused the request to add his name.

Without it, he does not lack this honor, but Liu Jiaxin lacks it.

With the addition of his name, many people in the mathematics community, not just the mathematics community, may feel that it is he, not the senior, who really solves the problem of factoring the positive integers.

But in fact, he knew very well that although he did help some, most of the things were made by Liu Jiaxin himself.

After coming back to his senses, Xu Chuan smiled and said, "No, P=NP? Conjecture is not within the scope of my research."

Hearing this, Terence Tao frowned slightly and asked, "But in that paper..."

Xu Chuan smiled and said: "I helped read the paper and made suggestions for improvement and flaws, and she revised it."

"So that's the case, no wonder." Tao Zhexuan suddenly understood, nodded, and continued: "I thought it was you who took action."

After a pause, he then asked: "Have you really not studied P=NP? The difficult problem? If you have, please tell me!"

Being suspected by Tao Zhexuan, Xu Chuan said with a bit of laughter and tears: "Just one Riemann Hypothesis is enough to consume all my energy. How can I have time to study P=NP? The difficult problem."

Opposite him, Terence Tao was stunned for a moment, a little confused.

"Wait a minute, are you studying the Riemann Hypothesis?"

Quickly coming back to his senses, he asked eagerly: "Is there any progress? How long will it take to solve it? Please be sure to send me the paper as soon as possible!"

Like Deligne, he asked almost without thinking, asking Xu Chuan to send him the paper as soon as possible.

Listening to Tao Zhexuan's request, the corners of Xu Chuan's mouth moved but he hesitated to speak.

Why do two people think he can solve the Riemann Hypothesis?

He doesn't even have the confidence himself, okay?