Time flies by.
In a blink of an eye, it's the end of October.
The weather has gradually turned cooler, and the falling autumn rains have also added a chill to the whole world.
Putting on his coat, Xiao Yi walked on the shore of Shushan Lake. This was what he had done most during this period.
However, this time he was the only one, and Luo Mingya has now devoted all his time to research.
Once they started researching, Luo Mingya basically put all his energy into it, just like he did when he was still in school. Therefore, it has been a while since they had much communication.
Somehow, it actually made him feel a little bit pity?
"Where does this emotion come from?"
He shook his head in his mind, and then stopped thinking about these messy things.
I continued walking along the path by the lake, occasionally seeing other people in the institute and saying hello to each other.
However, the main reason why he often runs to the lake now is because during this period of time, he has been thinking about what his next goal should be.
Theory, or applied technology?
This is a question worth wrestling with.
The former can satisfy oneself, while the latter has the opportunity to benefit all mankind.
However, studying the former, whether it is successful or not, can make you feel satisfied. After all, what is important is the research process. As for the results, the impact is not that big.
As for the latter, he was not interested in those ordinary problems, but the problems he was interested in were more complicated than the last. Therefore, it could only be said that there was a chance to solve it, but it was not certain that it could be solved.
"It seems like I'm a little too ambitious, right?"
Xiao Yi couldn't help but think about it in his heart.
Hmm, but having said that, the theoretical problem he plans to study is Riemann Hypothesis, so if we really want to say which problem is better for Gao Zhiyuan, it seems that we can't make a conclusion so easily.
If someone else were to look at it, it would probably be that every question is somewhat too ambitious.
"Well..."
"Forget it, it's nothing to worry about so much."
He raised his hand and looked at his watch.
"There is another number theory class this afternoon. Now you can get ready to go to school."
Maybe going to class or something can bring him some new thoughts?
Then he turned around and walked back. At the same time, he called Wang Li and asked him to prepare to pick him up and take him to school.
In the afternoon, USTC is in the classroom of freshman Hua Loo-Keng Class, School of Mathematics.
The classrooms are full, and students of all grades are 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, but these students still express great enthusiasm for Xiao Yi's class.
Even students who are about to graduate are still unwilling to be absent. Maybe there are students among them who have never missed any class of Xiao Yi.
Although Xiao Yi has said that those who attended his class last year should not come this year and leave their seats to other people in need, it still cannot stop the enthusiasm of these students.
Fortunately, after the school's coordination, the students who come to attend his classes now are basically from the School of Mathematics, which prevents students from other colleges from running over and taking up all the seats in the School of Mathematics.
In this way, the time came to 15:45.
Xiao Yi's figure appeared at the door of the classroom.
Some of the students in the classroom had already seen Xiao Yi in previous classes, while others had rarely gotten a seat and saw Xiao Yi for the first time.
But no matter what it was, he was very excited at this time.
After all, he is the idol in each of their hearts, and even the person they admire the most, so it is normal to feel excited every time they see him.
But Xiao Yi is basically not surprised by this.
These young people cannot correct themselves at all.
Um.
He is already 24 years old this year. To him, these students who have not graduated from a bachelor's degree are indeed considered young people.
He pressed his hand and signaled the students present to be quiet for a moment. There were still 10 minutes until the 15:55 class time. God knows whether these students would continue to noisy.
"There are still 10 minutes left in class. If you have any questions about the content of the previous class or when you go back to study on your own, you can come to me now to discuss it."
After saying the usual words, he sat on the chair on the podium and waited.
After a while, a group of students came towards him with scratch paper in their hands.
This link can be considered a necessary part of his class now.
This is also the reason why some students often come to his class even though they have already taken his number theory class, just for his Q&A session, especially since his answers are not limited to number theory questions, but can answer questions related to mathematics.
So much so that some graduate students did not even attend graduate classes, but came to attend his classes just to come to him with some problems they encountered in their thesis.
Among these graduate students, there are even doctoral students.
This also led some teachers to joke with Xiao Yi if they could pretend to be students and listen to his class, and then ask him questions during this session.
Just like that, almost twenty students ran to the podium and lined up, waiting to ask Xiao Yi questions.
Of course, the questions they ask are not particularly difficult, so basically Xiao Yi can give corresponding answers at a glance.
It took less than half a minute to ask one question, and within a short while, all the questions from more than 20 people had been answered.
Just then, the class bell rang, and this class officially began.
"Okay, classmates, let's start class."
"At the end of last class, we talked about some stuff related to prime numbers."
"I believe you all know that prime numbers are also the most important concept in number theory, and they involve many issues."
"And one of the most critical issues is the distribution of prime numbers."
"So, what we mainly talk about in this lesson is the distribution of prime numbers."
Xiao Yi turned around and wrote the words distribution of prime numbers 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 wondering about the application of this research, 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 laughed. This sentence was what Professor Xiao often said to them. The main purpose was to remind them that studying pure mathematics is not to make the results appear meaningful in practical applications.
Probably because students often asked him what role the study of number theory has in practical applications.
If it were some mathematicians with a bad temper like Perelman or Faltings, they would probably kick such students out without mercy.
Xiao Yi will say that in his various practical applications of mathematics in the past, this kind of pure mathematics has never been used.
After that, he would say something like this in class.
Turning around, he began to introduce science 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 already learned other aspects of prime numbers before, such as the infinity of prime numbers, Sieve of Eratosthenes, etc.
The current distribution of prime numbers is an overall application of the previous contents.
"... Then, what we are going to talk about at this time is, Prime Number Theorem."
"I think many of the classmates of Hua Loo-Keng Class know what Prime Number Theorem is. You may learn this when you participate in Math Olympiad."
"Prime Number Theorem describes the asymptotic distribution of prime numbers among positive integers. It is a landmark achievement in the process of studying the distribution of prime numbers in the mathematical community. It was independently proved by French mathematician Jacques Adamard and Belgian mathematician de la Vallebussan in 1896. Therefore, in the mathematical community, it is generally believed that Prime Number Theorem was jointly proved by these two mathematicians."
"Using 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 used a lot of the content in 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 Prime Number Theorem?"
Soon, students below raised their hands.
Xiao Yi remembers that this student was his freshman Hua Loo-Keng Class student.
"This classmate, please tell me."
The classmate quickly stood up and said very confidently, "I remember that the main knowledge they used was 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 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 normal points to him.
The student immediately sat down happily.
"Okay, so what I want to expand to everyone is Riemann zeta function."
"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 about the analytic continuation in complex analysis. This is also the most important knowledge point of 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 question this problem to obtain more useful information."
"..."
Xiao Yi is sometimes very pleased. The class he teaches is Hua Loo-Keng 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 the domain of definition was ignored. They thought that we could not discuss functions outside the domain of definition. They felt that this was meaningless."
"However, as your understanding of mathematics gradually deepens, you will understand this kind of problem. All you need to know now is that including 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 could easily begin to understand the process of analytical continuation based on their original concept of elliptic curves.
The difference is that the explanation method of Xiao Yi is a brand-new explanation of analytic continuation. It starts from elliptic curves and incorporates partial knowledge of modular form and L-function. Although most of the students present do not know what modular form and L-function are, 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 tell to some extent that in this method of Xiao Yi, a lot of things are used 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 a temporary thought of Xiao Yi.
However, it was not entirely accidental that he was able to think of this method.
Because it contains some of his recent thoughts on Riemann Hypothesis.
And Riemann Hypothesis is the ultimate problem of prime number distribution!
Until the end.
"... At this point, we have successfully transformed the domain of this ellipse."
"Now, we're starting to expand the domain of definition."
"At this point, we have achieved analytical continuation in another sense."
"Now, is there anything you don't understand?"
Xiao Yi Looking back, at this time, the students who were a little confused before basically understood.
As for what he hadn't figured out yet, he was helpless.
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 by no means a game for everyone.
After leaving some time for the students present to understand, he turned back and looked at the method he had temporarily given again, and suddenly fell into thinking.
When I was thinking smoothly just now, I hadn't discovered it yet.
But now after reading it again, he suddenly discovered that this new method of analyzing the process of analysis and extension seems to be a little different in it?
If he adds some Algebraic Geometry methods to this method...
Perhaps, he can directly transform the analytic continuation process of any analytic function into an elliptic curve Algebraic Geometry problem?
Or to put it directly, his main purpose is to directly transform the form of Riemann Hypothesis into such a question?
Inadvertently, a violent brainstorm began in his mind.
It's a pity that the students in the classroom will never think of their Professor Xiao. What they are thinking about at this time is how to solve Riemann Hypothesis.