Princeton University, Mathematics Department.
"Professor, I have encountered a little problem now... You see, I am now trying to use the method of analytic continuation to solve the problem of this function I am encountering now. I hope to obtain more information through analytic continuation."
"I've read a lot of related papers, and they generally do this in this step."
"But after I did this now, I didn't achieve the same effect as in other papers, and I didn't get the information I wanted."
In Manuel·Balgava's office, he was listening to the questions Sanjay asked him.
"Hmm... so what you are trying to study now is the properties of Gamma function?"
Manuel asked while looking at the scratch paper Sanjay handed him.
"Yes."
Sanjay nodded, "I mainly want to understand the behavior of this Gamma function on the entire complex plane, especially near some special points."
Manuel nodded.
Using the analytical continuation method to study Gamma function is indeed a very normal processing method. Most people will try to use this method to study Gamma function.
But...
Sanjay Why do you encounter difficulties at this point?
He looked at it carefully and began to analyze and extend this Gamma function in his mind based on commonly used methods.
First define it on the right half plane Re(z)>0: [Γ(z)=∫^∞ t^(z-1) e^(-t) dt]
Then, the property of Gamma function Γ(z+1)=zΓ(z) is used, allowing its definition to be extended to the entire complex plane.
Next...
Soon, he raised his eyebrows, and then smiled and said to Sanjay, "Oh! I think I know what problem you are encountering."
"The Gamma function you found contains some special properties, or in other words, some unexpected properties, and these unexpected properties cover up the information you want to obtain."
"Therefore, if you want to obtain this information through analytical continuation, my suggestion is that you can try to conduct a more in-depth analysis of the function after the continuation, and use other techniques, which may help you find some of the key information."
"Well, that's it."
After listening to the instructor's explanation, Sanjay was suddenly in a daze.
"Unexpected nature?"
"Do you need to use some other techniques?"
For a moment, he fell into thinking.
Seeing him thinking, Manuel gave a hint: "Actually, go read Elliptic Inverse Analytic Continuation's Elliptic Inverse Analytic Continuation paper by Xiao Yi, and then use this method to solve your problem with a high probability. Elliptic Inverse Analytic Continuation can specifically solve the problem you encountered."
Sanjay's eyes suddenly lit up, "Xiao Yi's paper? Okay! I'll read it later!"
Seeing Sanjay's expression, which seemed to be full of trust in his idol, Manuel shook his head with a smile.
Although these students of his are his students, they all admire Xiao Yi even more.
But he didn't say anything. Anyway, this kind of thing is too normal nowadays.
Just like in the last century, many mathematicians worshiped Grothendieck, but now, it's just that the object of worship has changed.
Even he himself used to be one of those who admired Grothendieck, but now, even though he is getting older, it is hard to say whether he admires or not, not to mention that Xiao Yi is younger than him.
But this does not prevent him from admiring Xiao Yi.
Mathematics is a field that really relies on strength. No fraudster can survive. As long as you have strength, you can gain the admiration of the entire mathematics community.
Xiao Yi has already gained the admiration of the entire mathematical community.
Soon, Sanjay returned to their laboratory and came to his work station.
The person sitting next to him was Chen Muhua, and he immediately said to Chen Muhua, "Hey, I just asked the professor about my problem, and he quickly found the problem and recommended that I study your former professor's Elliptic Inverse Analytic Continuation!"
Chen Muhua smiled and said, "It is indeed a good suggestion! Elliptic Inverse Analytic Continuation, it should be really helpful for the problem you encountered."
"Well! It finally gave me a reason to study Xiao Yi's paper." Sanjay said.
Then he opened the browser and started searching.
"Hmm... It seems that Xiao's papers are all posted on his personal website. Chen, quickly enter it for me. Xiao's personal website, so I don't have to look for it."
Chen Muhua shrugged and said, "I thought that as his loyal fan, you should know his website address very well."
Then he came to Sanjay and helped him enter the address.
Sanjay said, "Oh! Even loyal followers of India need to ask others for the way to Varanasi."
Chen Muhua rolled his eyes and didn't say much. He entered Xiao Yi's personal URL on Sanjay's website and finally pressed Enter.
"Okay."
Then he stood up and returned to his seat.
Sanjay then started looking for it. Although the website was in Chinese, the browser's built-in translation could still be used.
But soon, he frowned and asked, "Huh? Why didn't I find Elliptic Inverse Analytic Continuation?"
Chen Muhua was confused. "Isn't it just on the paper page?"
"I used the search function, but couldn't find it." Sanjay pointed at his computer screen and said.
Chen Muhua suddenly became even more strange.
Xiao Yi's personal website also comes with a very simple search function. However, although it is very simple, since the number of papers is not large, the search should be able to find it quickly.
There is absolutely no doubt about Professor Xiao's coding ability.
"Then, just look for it on the homepage."
Although he was confused, he then said.
If you can't find it, just look for it.
So he returned to the homepage and started to search for Sanjay.
But after searching for a while, his eyes widened.
He looked at a certain column on the homepage in shock, and it was directly marked with Riemann Hypothesis.
Obviously he didn't see it when he saw it before!
"Wait a minute, take a look at this. If I'm not mistaken, this thing is... the Riemann hypothesis? Take a look, isn't it?"
Sanjay was stunned for a moment, then took a look, and showed a surprised expression. "Yes... yes!"
He immediately guessed. "Could it be that Professor Xiao has also begun to announce his research progress?"
However, Chen Muhua didn't think so and said, "Let's go in and take a look first!"
Sanjay nodded immediately and then clicked on the link.
When you go in and take a look, the overall background is basically the same as Xiao Yi's personal website, and in the middle, there are five underlined links.
These are the names of five papers. Click on them to get the PDF files of these five papers.
The first paper is the paper of Elliptic Inverse Analytic Continuation that Sanjay was looking for just now.
And the second article...
That's what they can't understand.
"Discussion about High-Dimensional Modular Curve"
There is also a third... a fourth... and a fifth.
"Proof of Artin Conjecture"
"CM Elliptic Curve and Hecke feature"
"Proof of Riemann Hypothesis"
"A generous gift to Crab Te!"
After reading it, the two of them jumped up at the same time, hugged their heads, and exclaimed. And their actions naturally attracted the attention of others in the laboratory.
"Hey, what's going on with you two?"
Their shocked expressions quickly aroused doubts in others, and they started asking questions.
"Oh! You will never believe what we saw."
Sanjay said exaggeratedly.
"What is it?"
Others were even more curious.
At this time, a fellow student next to them walked over, took a look, and immediately jumped up in shock, even higher than them.
"I bought some fucking ones!"
Seeing the reaction of this third person, the other disciples immediately became more curious.
"Hey, hey, hey, you guys are not colluding to deceive us, are you?" Others said one after another.
Then someone called the fourth person next to them, "Jack, go and see what they are so shocked about!"
Jack stood up helplessly, and as he walked over, he also complained, "Hey, it's all boring. We are already Princeton University graduate students of the Mathematics Department, why don't we play these childish things... Wang Defa?!"
When he walked in front of the computer screen, he was similarly shocked.
"Hey, hey, hey! The four of you are already colluding, right?"
The others immediately said angrily.
At this time, others finally couldn't help it and walked over to see for themselves what was going on.
Of course, there is also one person who is convinced that they are simply being deceived and they can't help but watch it.
But after almost everyone looked at it with shocked expressions on their faces, this man finally hesitated, but he still didn't move and said, "Please, don't you all have such a tacit understanding, okay?"
However, at this time, the group of Princeton University mathematics graduate students gathered in front of the Sanjay computer had already started discussing.
"Is this true? Did he really prove it?"
"Since it was sent by him... then it must be true, right? After all, he is the smartest person in history!"
"I can't believe it, I really can't believe it... I haven't heard from him for so long. It turns out that he has actually solved this problem?"
"Oh my god, how long has it been? Is he really still a human? Why do I feel like he has already been reproductively isolated from ordinary humans?"
"But the question is... this is Riemann Hypothesis! Did he really prove it?"
After hearing this, the person who was still sitting on his seat finally understood what was shocking these people.
Xiao Yi, really proved Riemann Hypothesis?
He immediately shouted, "Don't lie to me! I really plan to come over now!"
But the group ignored him.
He stood up and began to walk over, and even the possible scenes that might appear next appeared in his mind. It was like the plot in those videos. A group of people were shocked by something, which caught the attention of the last person. As a result, that person walked over and took a look, only to realize that he was being played.
So, until finally, he walked over and took a look, and then he was shocked.
It's true!
"House! Why didn't you tell me earlier! This is actually true!"
But he ignored his excitement. After all, they were all in shock about this incident, and they had no time to pay attention to others' shock.
And maybe their voices were finally heard by Manuel in the office next door, so Manuel came over. When he saw a group of people gathered together, he frowned and said, "Hey hey hey, what are you doing? Noise is not allowed in my laboratory!"
Then, he got shocking news.
"Professor, Professor Xiao Yi... proved Riemann Hypothesis!"
His students all raised their heads to look at their respected mentor and informed them of the news.
These students all know very well that their teacher was one of the initiators of Prime Pioneer Program. After all, it was their teacher who was studying Xiao Yi with another professor Okounkov at the beginning. Elliptic Inverse Analytic Continuation attracted the attention of other professors. Finally, after discussion, this cooperation plan was decided.
The result is now...
They suddenly lost to that one person...
Manuel For the first time, I just didn't believe the news.
How is that possible!
That's Riemann Hypothesis!
How long has it been since?
No one in the world can solve Riemann Hypothesis so quickly!
But then, he suddenly realized that the other party was Xiao Yi.
Xiao Yi...
Standing there for a long time, this name lingered in my mind for a long time.
In the end, he didn't speak and just walked over. He wanted to know where these students of his got this news.
Soon, he came behind them and saw the page displayed on the computer screen.
Then he fell silent again.
He had the same reaction as the previous students.
This is actually true...
None of his students said anything, just looking at their professor.
They all wanted to know how their professor would react to this matter.
Until finally, they saw a kind of relief on the professor's face, as if to say: Yes, the other party is Xiao Yi!
His mood calmed down, and then he asked, "Where did you see this website?"
Sanjay said, "It's on Professor Xiao Yi's personal website. You know, you just recommended me to take a look at his Elliptic Inverse Analytic Continuation method. Then I went to this website to look at it and found this."
"That's right." Manuel nodded, then turned around and left without saying a word.
As the professor left, everyone in the laboratory immediately started talking.
Of course, their voices were a little quieter.
"Looking at it this way, all the big guys from Prime Pioneer Program have lost to Xiao Yi?"
"Please, it's not embarrassing, that's Xiao Yi!"
"That's the real GOD!"
"I will buy GOD."
At this time, someone suggested. "Let's open the paper and take a look first!"
This proposal was quickly approved, so Sanjay operated the computer and directly opened the last paper, which was "The Proof of Riemann Hypothesis" at the suggestion of everyone.
After clicking in and watching for a while, they fell silent.
At the level of these Princeton University mathematics graduate students, they found it a bit difficult to read this paper.
For a moment, they looked at each other.
"How about... let's forget it?"
"Forget it, forget it."
They all laughed, dispersed, and returned to their seats. It was still too difficult for them to understand this kind of paper.
However, this did not stop them from spreading the news online.
Riemann Hypothesis has been solved by Xiao Yi!
In Manuel's office, he first made a cup of coffee, then slowly sat in front of his computer and entered Xiao Yi's personal website.
Then, he quickly found the website.
Looking at the domain name of this website, he sighed helplessly.
This domain name is completely suitable for the content.
He then turned his attention to five papers.
After briefly looking at their names, "Discussion on High-Dimensional Modular Curve" and "Proof of Artin Conjecture" quickly attracted his full attention.
"High-Dimensional Modular Curve? So his most important method is to develop a high-dimensional case of the modular curve?"
"And then there's this...Artin Conjecture..."
He sighed, "He actually chose the path of Artin Conjecture."
And this path, among their Prime Pioneer Program, was almost abandoned immediately.
"Okay, let's start watching."
Of course, before that, he did not forget to share this website to their Prime Pioneer Program discussion group, and attached a message: [Xiao Yi has completed the proof of Riemann Hypothesis.]
Then, he stopped paying attention to the question marks that kept coming back and started reading the paper.