"Princeton University and the Annals of Mathematics would like to announce that Dr. Qiao Yu's paper "Proof of the Riemann Hypothesis Based on the Generalized Modal Axiom System" has completed rigorous peer review and has been accepted for publication.

After two months of in-depth review, a review team composed of 12 top international mathematicians and feedback from relevant academic networks confirmed that the proof proposed in the paper is completely correct and self-consistent.

This paper provides a solution to the Riemann Hypothesis, a core problem in mathematics that has lasted for nearly two centuries, marking a milestone achievement in the history of mathematics.

Dr. Qiao Yu's thesis proposed a new mathematical framework that spans the boundaries of traditional number theory and logic. It uses the generalized modal axiom system to construct a formal semantics and logically proves the symmetry of the non-trivial zero points of the Riemann zeta function.

and distribution rules.

The following are the main contributions of the paper..."

That night, the official website of Princeton University finally updated the announcement.

The title of the announcement is also very grand - "Official Statement on Dr. Qiao Yu's Paper Solving the Riemann Hypothesis".

This statement not only announced the official publication date of the paper, but also proposed plans for a special issue. It also included comments from three reviewers.

Tao Xuanzhi: "Dr. Qiao Yu's work pioneered the introduction of modal logic into number theory. This is not only a proof of the Riemann Hypothesis, but also a methodological revolution."

Peter Schulz: "This paper shows the perfect combination of geometry, number theory and logic, opening up a new path for mathematical research."

Akushel Betes: "The proof of the Riemann Hypothesis not only solves a century-old problem, but also adds an extremely important weapon to the number theory toolbox - generalized modal logic."

There is no doubt that these are already very high evaluations. Of course, this paper also deserves this evaluation.

In fact, no matter who solves the Riemann Hypothesis, he will probably get similar evaluations - just mindless bragging.

As Princeton University commented in a statement on its official website, this is a problem that has troubled the world's top mathematicians for nearly two centuries.

What's more, this conjecture is still so important.

The ζ function formula is still essentially the sum of an infinite series, so there is a convergence problem. For any complex number with a real part greater than 1, the sum is convergent.

Then, mathematical techniques are used to extend the domain of the ζ function to the non-convergent region.

The ζ function has infinite non-trivial zero points, and no graph can express all of these non-trivial zero points.

More than a hundred years ago, the mathematician Riemann believed that the real parts of these non-trivial zero points are on the straight line 1/2 of the complex plane.

In fact, so far, mathematicians have verified more than 10 billion non-trivial zero points through calculations, and the real parts of these non-trivial zero points are indeed on the 1/2 straight line, without exception.

But it is a pity that limited verification is not equivalent to proof at the mathematical level. After all, for the requirements of mathematical rigor, facing the concept of infinity, whether it is verified 10 billion, 20 billion or more, it cannot replace it.

The proof process of mathematical logic.

After all, theoretical mathematics is a subject that can completely overthrow the existing number theory framework with just one counterexample.

Today this problem has finally been proven!

Of course, proving the Riemann Hypothesis is never just about verifying the correctness of this conclusion. The most important thing is to reveal why this conjecture is correct. As the three reviewers commented, the most important significance of Qiao Yu's paper is that it provides

method!

The applicability of the generalized modal axiom system has been verified again. Even if these mathematicians do not yet know the application of this axiom system in computational mathematics, what is certain now is that this will be an indispensable toolbox for future number theory research.

!

Having this toolbox means that a path has been opened up for everyone, and many number theory problems may be able to be solved easily!

So, on this evening in China, when Princeton posted this statement on its official website, the world of mathematics exploded!

This time it was really explosive!

Even those mathematicians who were not interested in the Riemann Hypothesis before have mostly read Qiao Yu's paper and are paying attention to the opinions given by these reviewers.

After all, this reviewer lineup is already strong enough. If these people are sure that there is no problem with Qiao Yu's argumentation process, then there must be no problem.

Not to mention that this time it was an open community verification, and Princeton University officially announced the results.

Qiao Yu is still in his teens, not even an adult yet. He still has more than 20 years of prime time in mathematical research. To be honest, this is scary...

Really, after more than twenty years, God knows how far Qiao Yu can grow.

Suddenly, after Princeton issued the official announcement, the mobile phones of mathematicians around the world began to become busy.

Although 2026 is only halfway through, there is no doubt that this is already the hottest topic in mathematics this year.

…

"Pierre, do you really think that child's certificate is flawless? So you gave it all a pass?"

"No, Lucas, there is actually a small loophole. Almost everyone analyzed it word for word and found it, but it only took him a week to fix that small loophole.

You will know when the official version of the paper is published. In addition, he also proposed three continuity conjectures. Believe me, they are all very valuable conjectures. If they can all be solved, we will have a new understanding of the distribution of prime numbers."

The other party was silent for a moment and continued: "Okay. It seems that we need one more candidate for the Fields Medal this year."

"Haha, this is inevitable! Anyone who solves the Riemann Hypothesis will get a Fields Medal. Even if he is over forty years old, he will get a silver medal, right? What's more, Qiao Yu is only seventeen years old

!Damn it, he is a once-in-a-millennium genius!"

"Yes, you are right! He is only seventeen years old! Hey, every time I think about his age, I feel very emotional. Pierre, we are all old. There is not much time left for us."

"Okay, Lucas, stop feeling emotional! Do you know why I am in a good mood today? At least before we die, someone has proved the Riemann Hypothesis! Don't you think this is our luck?

I just finished a phone call with Professor Yuan, and he told me an old Chinese saying: Hear the Tao in the morning and die at night. It means that if you can understand the truth in the morning, you will be satisfied even if you die at night.

That's what he thinks now!

So no matter what, we are much luckier than Sir Atiyah. He held the last report meeting in his life three months before his death, but unfortunately his report was not recognized."

Lucas Eisen sighed inwardly when he heard these words.

From this point of view, Pierre is indeed much luckier than Atiyah. This British knight is still working hard to prove the Riemann Hypothesis at the age of 89.

But it is really a pity. The proof process he gave can only be said to be difficult for many colleagues to evaluate. This is indeed the case. Many people, including his colleagues, refused to publicly evaluate this achievement.

He even commented privately that the old Sir may have been really old and confused in the last stage of his life, and was deceived by the researchers who wanted to get resources from him, so that he talked nonsense in the last stage of his life.

This is really regrettable.

Lucas Eisen also knew why Pierre specifically mentioned Michael Atiyah.

After all, Pierre Delini was Alexander Grothendieck's most famous student, and Atiyah's early research was also greatly influenced by Grothendieck.

For example, Atiya's earliest achievement was to co-create topological K theory.

From this point of view, Pierre is obviously much luckier than Atiyah. At least he witnessed the Riemann Hypothesis being solved with his own eyes, and was even one of the reviewers of this paper.

Of course, this also means that it is a certainty that Qiao Yu will win the Fields Medal this year. Originally, Lucas thought that Qiao Yu should accompany him this year and win the award at the next World Congress of Mathematicians.

But now it seems impossible! After all, no one would want to offend the twelve reviewers, not even him.

After a few more brief conversations, Lucas Eisen hung up the phone. After sitting in the office and thinking for a few minutes, he suddenly thought of his grandson-in-law Frank.

Really, he originally thought Frank was also a very talented guy.

When Frank was still in love with his granddaughter, the paper he published, "From Modular Forms to Modular Spaces: Hecke Algebraic Actions on Algebraic Curves," once made his eyes light up.

Later, he published "Hecke Algebraic Properties and Applications on Algebraic Curves".

After the two young people got married, Lucas Eisen really trained Frank as his successor. He hoped that Frank could succeed him and become an academic leader in the intersection of algebraic geometry and number theory at Berkeley in the future.

So he introduced Frank to the research group on the Geometric Langlands Conjecture. At that time, Frank's performance was actually not bad. In 2021, he also published "Analysis of Low-Dimensional Curves in the Geometric Langlands Framework".

This paper proposed the geometric Langlands correspondence method on low-dimensional curves, which simplified the processing of traditional high-dimensional module spaces. He also established a foothold at Berkeley, teaching "Introduction to Algebraic Geometry".

But what dissatisfied him was that since Qiao Yu found fault with his paper, this guy's mathematical spirituality seemed to have suddenly disappeared.

It's just that for such a long time, there has been no tangible results at all. He suggested that Frank contact Qiao Yu to conduct research on related topics together, but the result was also messed up.

This reminded Lucas of the algebraic geometry conference where he specially called Frank over and told him in earnest that he needed to have a good relationship with a rising math star like Qiao Yu, which was a waste of all his efforts.

I have been confused day by day, and I still haven’t found a research direction, which is really disappointing.

Well, actually this can’t all be blamed on Frank. After all, there are so many people who want to cooperate with Qiao Yu. And it’s actually quite difficult to find a new research direction in more than a year.

But what can I say? What people fear most is comparison.

After Qiao Yu proved the geometric Langlands conjecture, he did a lot more and proposed a generalized modal axiom system, reducing the interval between prime pairs to 6, and now he has directly proved the Riemann conjecture.

Frank has accomplished nothing so far, and can even be said to have no direction. The contrast is obvious. If you add that both of them are Chinese, the contrast becomes even more obvious.

There is no doubt that Frank did not show any temperament or ability to become an academic leader during this period.

This also made Lucas feel very dissatisfied, after all, he invested a lot of energy and resources in this guy.

After sulking for a while, Lucas Eisen did not calm down. Instead, he became more and more angry as he thought about it. Then he simply took out his cell phone and called Frank.

"Hey, Frank, you should have seen the statement just released by Princeton University, right?"

"Yes, Lucas, I saw it. It's unbelievable that he really solved the Riemann Hypothesis. In fact, I have been studying his paper these days, and I have some insights."

Lucas Eisen felt that the other party's tone was very depressed, which made him feel even more angry. Do you have any experience reading the paper now? What have you been doing?!

"I just had a phone call with Pierre, you know? After two months of discussion, all twelve reviewers and the team behind them feel that there is no problem with Qiao Yu's final proof! You should understand what this means.

What!"

Looking at what he said, Frank on the other end of the phone felt that anyone in the world could understand what this meant. Especially when the World Congress of Mathematicians will be held next month.

"Of course, Lucas, I understand. It is impossible for the Fields Medal jury to ignore the Riemann Hypothesis. Oh, no, it should be the status of Riemann's theorem now. So Qiao Yu will be able to win the Fields Medal this year.

,Is it right?"

"Yes, it seems that you are still sober, so now you understand why I asked you to try your best to promote cooperation with Joe? But look what you did?

You wasted a once-in-a-lifetime opportunity. I told you to take advantage of the fact that you and Qiao are both Chinese and build a good relationship with him. I don't understand where your inexplicable sense of superiority comes from.

Is it because he is young that makes you feel embarrassed? If that is the reason, then you are too stupid! Really, I have always thought that you are a smart person who knows what is important and what is light!"

Lucas Eisen's sermon left Frank speechless.

He wanted to say something to Lucas Eisen, "Because I am Qiao Yu's father! And he doesn't even know about it yet!" But he still didn't say it out loud.

Frank was not worried about those romantic affairs in his youth being known by Lucas. In fact, with his current status, it was not a big problem for such things to be exposed.

The main reason was that the way it was handled at the time was inappropriate, and it felt a bit like abandoning his wife and children. Lucas is the kind of person who prides himself on being very family-oriented.

The most important thing is that from the perspective of Chinese culture, his behavior at that time was not right. If he recognized Qiao Yu rashly, he was afraid of turning against Qiao Yu.

In fact, ever since Qiao Yu announced on arXiv that he had proved the Riemann Hypothesis, he has been suffering every day recently.

And the mentality is extremely complex.

On the one hand, he hopes that Qiao Yu is right. No matter what, he has confirmed that Qiao Yu is his biological son.

On the other hand, he didn't want Qiao Yu to be so powerful. Frank was really afraid that Qiao Yu might take revenge on him if he knew what happened back then.

After all, Qiao Yu has been living with his mother all these years. Frank doesn't know what Qiao Xi said about him to Qiao Yu, but presumably Qiao Xi doesn't say anything good about him.

Being blocked by a mathematical genius who solved the Riemann Hypothesis at the age of seventeen, Frank felt that he would probably have a bleak future in mathematics. After all, he really knew how powerful academics could be in this era.

Qiao Yu happens to have all the elements needed to become a generational scholar. Damn it, is this going to keep people alive?

At this time, I have to face Lucas’ merciless questioning...

Frank regrets it! Very regrets it!

If he had not chosen to go abroad, even if he had returned to China to find Qiao Xi after graduating from MIT, he would have admitted his mistake and assumed the responsibility of raising Qiao Yu, and raised Qiao Yu by himself...

Just thinking about how bright the future would have been if he had chosen this way, Frank wanted to slap himself twice!

God originally gave him a shortcut to the sky, but in the end he chose the road behind the shortcut.

It's a pity that there are no what-ifs in life, so now he is speechless when faced with Lucas' question.

Of course, even if he had something to say, he wouldn't say it. Frank knew the character of Lucas Eisen, he was a willful old man.

If he opened his mouth to defend, it would only make the old man angrier and make a lot of uglier words. But he had no choice now and had to rely on Lucas to pave the way for him. So he could only endure this scolding.

"I'm sorry, Lucas, I'm disappointed. But as you know, there are too many people who want to cooperate with Qiao Yu. It's not that I didn't try, but he rejected me."

"Hmph...you can do it on your own! If you don't even know how to use your own advantages, how can you expect me to help you?"

After saying that, Lucas Eisen hung up the phone angrily.

On the other side, Frank listened to the busy tone on the phone, his face uncertain.

There are many things to consider, such as how should he face Qiao Yu at the upcoming World Congress of Mathematicians?

Especially when he heard that Qiao Xi was now accepted by Yuan Zhengxin, and he didn't know if Qiao Xi would go to the World Congress of Mathematicians. If he went there, if he met him and recognized him...

But it was impossible not to attend this meeting. After all, although he was not qualified to give a full report in this meeting, it was determined that he would give a thirty-minute report in a branch room, and he could not give up this opportunity.

After all, even if he doesn't win the Fields Medal, he still has the Abel Prize and the Wolf Prize to look forward to in the future, so he can't help but go to such a top conference every four years to show his face, right?

Sad!

…

Huaxia, Lu Yanxiang of the Institute of Mathematics and Systems Science of the Chinese Academy of Sciences is also worried.

Of course, he was not as worried as Frank. But he still felt distressed after seeing Princeton’s announcement...

It’s really recognized!

It can only be said that Zhang Yuanling is in good company. But Lu Yanxiang really wanted to find someone to talk to him. Qiao Yu directly applied for a project to solve such a big problem as the Riemann Hypothesis. Is there any problem with him rejecting it?!

What the hell is wrong with this?!

Before the Riemann Hypothesis was solved, who could believe that a teenager could really solve this problem and leave a super luxurious review team speechless?

The most annoying thing was that when he called Zhang Yuanling to complain, the guy actually said: "Hey... Academician Lu, our situations are completely different. We can't be confused.

After all, Qiao Yu withdrew his project by himself at that time. I just asked a few questions and didn’t mean to deny it. But your situation is different. You directly rejected his project!"
Chapter completed!