Section 335

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

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

Although Xiao Yi has said that those who have listened to his class last year should not come this year, leaving the seats for other people with more needs, it still can't stop the enthusiasm of these students.

Fortunately, after the school's coordination at the beginning, the students who come to listen to his class now are basically all from the School of Mathematics, avoiding students from other colleges running over and taking up all the seats in the School of Mathematics.

In this way, the time has come to 15:45.

Xiao Yi's figure appeared at the door of the classroom.

Some of the students in the classroom have seen Xiao Yi in previous classes, while others have rarely grabbed a seat and are seeing Xiao Yi for the first time.

But no matter which one, they are very excited at this time.

After all, he is the idol in everyone's mind, even the person they admire the most. It is normal to feel excited every time they see him.

Xiao Yi is basically used to it.

These young people, it is impossible to correct them.

Yeah.

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

He pressed his hand and signaled the students present to be quiet for a while. There are still 10 minutes before the class time at 15:55. God knows whether these students will continue to quarrel.

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

He said as usual, and then he sat on the chair on the podium and waited.

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

This link is also a necessary link in his class now.

This is also the reason why some students, even though they have already taken his number theory class, still often come to his class, just for his Q&A session, especially because his answers are not limited to number theory questions, but can answer questions related to mathematics.

So much so that some graduate students do not even take graduate classes, but come to his class just to ask him questions encountered in their papers.

Among these graduate students, there are even doctoral students.

This also made some teachers joke with Xiao Yi, saying that they could also pretend to be students to listen to his class, and then ask him questions in this session.

In this way, almost more than 20 students ran to the podium, lined up, waiting to ask Xiao Yi questions.

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

It took less than half a minute to answer a question, and just like that, after a while, all the questions of more than 20 people were answered.

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

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

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

"I believe you all know that prime numbers are also the most important concept in number theory, and it involves many problems."

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

"So, what we mainly talk about in this class is the distribution of prime numbers."

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

"So, at this time we have to return to a very fundamental question, why do we study the distribution of prime numbers?"

"Ahem, of course, if you are still struggling with the role of the study of this problem in application, then I still want to remind you not to think about this kind of question that is destined to have no answer."

The students present all smiled. This sentence is also what Professor Xiao often said to them. The main purpose is to remind them that studying pure mathematics is not to make the results seem meaningful in practical applications.

It is probably because students often asked him in the past what role the study of number theory has in practical applications.

If it were Perelman or some other mathematicians with a bad temper, they would probably kick such students out without mercy.

Xiao Yi would say that he had never used this kind of pure mathematics in various practical applications of mathematics in the past.

Then, he would say this in class.

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

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

They had already learned other aspects of prime numbers, such as the infinity of prime numbers, the sieve of Ehrlich, and so on.

The current prime number distribution is a comprehensive application of the previous content.

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

"I think many of the students in Hua Luogeng's class know what the prime number theorem is. You may learn this when you participate in the mathematics competition."

"The prime number theorem describes the asymptotic distribution of prime numbers among positive integers. It is a landmark achievement in the mathematical community in the process of studying the distribution of prime numbers. It was developed in the 18th century by Adama and the Belgian mathematician de la Vallebe. Sang gave proofs independently one after another, so in the mathematical community, it is generally believed that the prime number theorem was jointly proved by these two mathematicians.”

"Using the prime number theorem, we can give a very approximate distribution of prime numbers and get a lot of information from it, such as the Elliott-Halberstam conjecture that I once proved, which uses a lot of the content of the prime number theorem."

Xiao Yi said: "Here, let's expand a little bit. Do you know what knowledge Jacques Adama and de la Vallebusan mainly used when they proved the prime number theorem?"

Soon, students below raised their hands.

Xiao Yi remembered that this student was a student in his freshman Hua Luogeng class.

"This classmate, please tell me."

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

Xiao Yi nodded with satisfaction: "Yes, I can see that you do have a relatively deep understanding of this aspect."

Then he asked the student's name and said he would add some regular points to him.

The student immediately sat down happily.

"Okay, so what I want to expand for you 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 this will also be a more important field, so I will take this opportunity to tell you in advance. Let’s talk about the analytic continuation in complex analysis, which is also the most important knowledge point about the Riemann zeta function.”

"The so-called analytic continuation means that we artificially change the domain of the analytic function, extending the original smaller domain to a larger domain, and then let us solve the problem to get more Useful information.”

"..."

Xiao Yi is sometimes very pleased. The class he teaches is Hua Luogeng's class, so even if what he teaches is difficult, these students can accept it, and there is a high probability that they will continue to study independently after returning.

In this way, the method of analyzing continuation was explained, and most of the students in front of them quickly understood this method.

Xiao Yi also briefly demonstrated how to use analytic continuation to prove how 1+2+3+4+... is equal to -1/12.

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

"The so-called analytic continuation allows us to ignore the boundaries of the domain of definition."

"Maybe some students were unable to turn around for a while. They thought why we just ignored the domain of definition. They thought we couldn't discuss functions outside the domain of definition. They thought it was meaningless."

"However, as your understanding of mathematics gradually deepens for this kind of problem, you will understand that all you need to know now is that including the Riemann Hypothesis, it is based on this method."

"But, in order for you to understand, I will explain it to you from another angle using elliptic curves."

Xiao Yi turned around and started writing on the blackboard.

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

Elliptic curves are something that has been learned in high school mathematics. The freshmen here who have just entered school for less than two months naturally still remember elliptic curves.

Under Xiao Yi's explanation, they were easily able to gradually begin to understand the process of analytical continuation based on their original concept of elliptic curves.

The difference is that Xiao Yi's explanation is a brand-new explanation of analytic continuation. It starts from elliptic curves and incorporates partial knowledge of modular forms and L functions. Although most of the students present do not know They know what modular form and L function are, but because Xiao Yi's explanation only contains part of the knowledge, it is not difficult for them to understand it.

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

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

Analyzing continuation, can it still be understood from this perspective?

Although they can't see it clearly, they can know to some extent that Xiao Yi's method uses a lot of things comprehensively. The only thing they can see is the model form.

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

"Professor Xiao is really well-intentioned in order to give a good class. He was able to come up with such a method to explain analytic continuation in advance."

However, they didn't know that in fact, this method was just Xiao Yi's temporary thought.

However, it was not entirely accidental that he was able to think of this method.

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

And the Riemann hypothesis is the ultimate problem of prime number distribution!

Until the end.

"... here, we have successfully transformed the domain of this ellipse."

"Now, we start to expand the domain."

"Here, we have also achieved analytical extension in another sense."

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

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

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

Although many mathematicians have said that mathematics is not just a game for geniuses, he sometimes adds a sentence at the end, but it will never be a game for everyone.