The top student must be diligent.

Chapter 273 Another Implementation of Parsing Extension

Time flies by.

In the blink of an eye, it was the end of October.

The weather has gradually turned cooler. Each autumn rain brings a chill. As the autumn rain falls, the whole world becomes colder.

Putting on his coat, Xiao Yi walked along the shore of Shushan Lake. This was the thing he did most often in recent times.

However, this time he was alone, and Luo Mingya had now devoted all his time to research.

Once she started the research, Luo Mingya basically put all her energy into it, just like when she was still in school. Therefore, they had not had much communication for a while.

Somehow, he actually felt a little bit regretful?
“Where does this emotion come from?”

He shook his head in his heart and stopped thinking about these messy things.

I continued walking along the path by the lake, occasionally seeing other people from the institute and we would say hello to each other.

However, the reason why he often runs to the lake now is mainly because he has been thinking about what his next goal should be during this period of time.

Theory or applied technology?
This is a very worthy issue to ponder.

The former can satisfy oneself, while the latter has the opportunity to benefit all mankind.

However, researching the former, regardless of whether it succeeds or not, can make one feel satisfied. After all, what is important is the research process. As for the results, the impact is actually not that great.

As for the latter, he was not interested in those ordinary problems, but the problems he was interested in were more complicated than each other, so it could only be said that there was a chance to solve them, but it was not certain that they could be solved.

"So, am I a little too ambitious?"

Xiao Yi couldn't help but think about this in his mind.

Well, but then again, the theoretical problem he plans to study is the Riemann hypothesis, so if we really want to say which problem is more ambitious, it seems that we can't make a conclusion so easily.

If someone else were to look at it from their perspective, they would probably think that each question was a bit too ambitious.

"Um……"

"Forget it. It's no use fretting so much."

He raised his hand and looked at his watch.

"I have another number theory class this afternoon, so I can get ready to go to school now."

Maybe attending classes or something could give him some new ideas?

Then he turned around and walked back, and at the same time called Wang Li, asking him to prepare to pick him up to go to school.

……

In the afternoon, in the classroom of the freshman Hua Luogeng class at the School of Mathematics of USTC.

The classroom is packed with students from all grades waiting for their teachers to arrive.

Although so many years have passed, the students in the school are basically very familiar with their star teacher. However, these students still show great enthusiasm for Xiao Yi's class.

Even students who are about to graduate are still reluctant to be absent. Maybe, there are students among them who have never missed any of Xiao Yi's classes.

Although Xiao Yi has already stated that those who attended his class last year should not come this year and leave their seats for other people in need, it still cannot stop the enthusiasm of these students.

Fortunately, after coordination with the school, the students who come to his class now are basically all from the School of Mathematics, which avoids students from other colleges coming over and taking up all the seats in the School of Mathematics.

In this way, the time came to 15:45.

Xiao Yi appeared at the door of the classroom.

Some of the students in the classroom had seen Xiao Yi in previous classes, while others had the rare opportunity to grab a seat and this was their first time seeing Xiao Yi.

But no matter which one it was, they were all very excited at this moment.

After all, he is the idol in each of their hearts, even the person they admire most, so it is normal to feel excited every time they see him.

Xiao Yi was basically not surprised by this.

These young people are totally beyond correction.

Ok.

He is already 24 years old this year, and to him, these students who have not even graduated from college are indeed considered young people.

He raised his hand, signaling the students present to be quiet. There were still 15 minutes before the class at 55:10, and God knew whether these students would continue to make noise.

"There are 10 minutes left before class. If you have any questions about the content of the last class or during your self-study, you can come and talk to me now."

He said the usual thing, then sat down in the chair on the podium and waited.

After a while, a group of students came towards him with draft papers in their hands.

This link is now a necessary part of his class.

This is also the reason why some students still come to his class frequently even if they have already taken his number theory class, just for his question-and-answer session, especially because his answers are not limited to number theory questions, but can answer any questions related to mathematics.

To the extent that some graduate students did not even attend graduate classes, but came solely to attend his classes, just to come to him with problems they encountered in their papers.

Among these graduate students, there are even doctoral students.

This also led some teachers to joke with Xiao Yi, asking if they could also pretend to be students and listen to his lectures, and then ask him questions during this session.

Just like that, nearly twenty students ran to the podium, lined up, and waited to ask Xiao Yi questions.

Of course, the questions they asked were not particularly difficult, so Xiao Yi could basically give corresponding answers at a glance.

Each question took less than half a minute, and in a short while, all the questions from more than 20 people were answered.

Just then, the bell for class rang, and the class officially began.

"Okay, students, let's start the class."

"At the end of last class, we talked about something related to prime numbers."

"I believe you all know that prime numbers are also the most important concept in number theory. They involve many problems."

"One of the most critical issues is the distribution of prime numbers."

"So, what we're going to focus on in this class is the distribution of prime numbers."

Xiao Yi turned around and wrote the four words "prime number distribution" on the blackboard.

"Well, now we have to return to a very fundamental question: why do we want to study the distribution of prime numbers?"

"Ahem, of course, if you are still wondering what role the research on this problem will play in terms of application, then I still want to remind you not to think about such a question that is destined to have no answer."

The students present all smiled. This sentence was something Professor Xiao often said to them. The main purpose was to remind them that the study of pure mathematics was not to make the results have some significance in practical applications. It was probably because students often asked him what role the study of number theory played in practical applications.

If it were a mathematician with a more explosive temper like Perelman or Faltings, they would probably kick such a student out without mercy.

Xiao Yi would say that in his previous various practical applications of mathematics, he had never used this kind of pure mathematics.

After that, he would say something like this in class.

Turning around, he began to explain to the students present the history of research on prime number distribution and the origins of various related theories.

This can also be regarded as a review of some of the content they have learned before.

They have previously learned other aspects of prime numbers, such as the infinity of prime numbers and the Sieve of Ehrlich.

The current distribution of prime numbers is a comprehensive application of the previous contents.

"...Then, what we are going to talk about at this time is the prime number theorem."

"I think that many of the students in Hua Luogeng's class should know what the prime number theorem is. You may learn about it when you participate in the math competition."

"The prime number theorem describes the asymptotic distribution of prime numbers among positive integers. It is a milestone achievement in the study of prime number distribution in mathematics. It was independently proved by French mathematician Jacques Hadamard and Belgian mathematician de la Vallee-Boussan in 1896. Therefore, it is generally believed in the mathematics community that the prime number theorem was proved by these two mathematicians together."

"Using the prime number theorem, we can give a very approximate approximate distribution of prime numbers and get a lot of information from it. For example, the Elliott-Halberstam conjecture that I proved used a lot of information from the prime number theorem."

Xiao Yi said: "Let's expand on this a little bit. Do you know what knowledge Jacques Adama and de la Vallee-Boussan used when they proved the prime number theorem?"

Soon, some students raised their hands.

Xiao Yi remembered that this student was from the freshman Hua Luogeng class he taught.

"Student, please tell me about this."

The student stood up quickly and said confidently: "I remember that the main knowledge they used was the Riemann zeta function given by Riemann. The key step was to prove that if the complex number s can be written in the form of 1+it, and t is greater than 0, then zeta (s) ≠ 0."

Xiao Yi nodded with satisfaction: "Not bad, it can be seen that you do have a relatively deep understanding of this aspect."

Then he asked the student's name and said he would give him some extra points.

The student immediately sat down happily.

"Okay, so what I want to expand on is the Riemann zeta function."

"The Riemann zeta function involves the method of complex analysis. As for complex analysis, you can also learn it later, and it will also be a relatively important field. So I will take this opportunity to tell you in advance about the analytic extension in complex analysis, which is also the most important knowledge point of the Riemann zeta function."

"The so-called analytical extension means that we artificially change the domain of the analytical function, expand the original smaller domain to a larger domain, and then let us solve the problem again to obtain more useful information."

"..."

Xiao Yi sometimes feels very gratified because the class he teaches is Hua Luogeng's class. Therefore, even if what he teaches is difficult, these students can accept it, and there is a high probability that they will study independently after returning home.

In this way, the method of analytical extension was explained, and most of the students in front of me quickly understood this method.

萧易还简单展示了一下，如何利用解析延拓来证明1+2+3+4+……是怎么等于-1/12的。

However, seeing that there were still some students who did not understand, Xiao Yi thought for a moment and then said, "Then next, I will show you a method that is easier to understand."

"The so-called analytic extension is to let us ignore the boundaries of the domain of definition."

"Some students may find it difficult to understand why we have to ignore the domain of definition. They think we cannot discuss functions outside the domain of definition and think it is meaningless."

"However, as your understanding of mathematics deepens, you will understand this kind of problem. Now you only need to know that the Riemann hypothesis is based on this method."

"But, in order to make you understand, I will use the elliptic curve method to explain it to you from another perspective."

Xiao Yi turned around and started writing on the blackboard.

"We first give an ellipse equation, which we simply express as y^2=x^3+ax+b, where a and b are real numbers."

Elliptic curves are something we learned in high school mathematics. These freshmen who have just been enrolled for less than two months naturally still remember elliptic curves.

With Xiao Yi's explanation, they were able to easily begin to understand the process of analytical extension based on their original concepts of elliptic curves.

The difference is that Xiao Yi's explanation method is a completely new explanation of analytic extension. Starting from elliptic curves, it incorporates some knowledge of modular forms and L-functions. Although most of the students present did not know what modular forms and L-functions were, because Xiao Yi's explanation only included part of the knowledge, it was not difficult for them to understand.

As for the other undergraduates, they felt a little confused and a little impressed.

However, for several graduate students in the classroom, they were a bit shocked.

Can analytical extension be understood from this perspective?
Although they couldn't see it clearly, they could more or less know that Xiao Yi's method combined many things into one. All they could see was the model form.

For a moment, they couldn't help but sigh.

"Professor Xiao really put a lot of effort into teaching a good class. He even came up with a method to explain the extension of analysis in advance."

However, they didn't know that, in fact, this method was just something Xiao Yi thought of at the last minute.

However, it was not entirely accidental that he came up with this method.

Because it contains some of his recent thoughts on the Riemann hypothesis.

The Riemann hypothesis is the ultimate problem of prime number distribution!
till the end.

"…At this point, we have successfully transformed the domain of the ellipse."

“Now, we’re starting to expand the domain.”

"At this point, we have achieved analytical extension in another sense."

"Now, is there anything you don't understand?"

Xiao Yi turned around, and at this time, the students who were a little confused before basically understood.

As for those who haven't figured it out yet, he can't help them.

Although many mathematicians have said that mathematics is not just a game for geniuses, he sometimes adds a sentence at the end, but it is definitely not a game for everyone.

He gave the students present some time to understand the method, then turned around and looked again at the method he had temporarily given, and suddenly fell into deep thought.

When I was thinking smoothly just now, I hadn't discovered it yet.

But now after reading it again, he suddenly found that this new method of analyzing the process of analytical extension seemed to be a little different?
If he added some algebraic geometry methods to this method...

Perhaps, he could transform the analytical extension process of any analytical function directly into an algebraic geometry problem in elliptic curves?

Or to put it simply, his main purpose is to directly transform the form of the Riemann hypothesis into such a problem?
Before he knew it, a violent brainstorm began to stir in his mind.

It’s a pity that the students in the classroom would never think that their Professor Xiao was thinking about how to solve the Riemann hypothesis at this moment.

……

(End of this chapter)