The top student must be diligent

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 algebraic geometry problem on elliptic curves?

Or to put it bluntly, his main purpose is to directly transform the form of the 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 the Riemann Hypothesis.

…

Chapter 274 The Road to Proving the Riemann Hypothesis

There wasn't much time to think, after all, he still had to go to class, so Xiao Yi quickly stopped his thinking, turned to look at the students present, and said: "Okay, classmates, let's continue talking. , how to use this analytical continuation method to obtain further results on the distribution of prime numbers."

"Here, we are going to explain what the Riemann zeta function is."

Then, Xiao Yi began to write on the blackboard.

From the Riemann zeta function, combined with the prime number theorem, the form of the Riemann conjecture is finally revealed, and then it is explained how much the Riemann sub-conjecture is related to the distribution of prime numbers.

"Some students may think that as long as the Riemann Hypothesis is proved, we can find the general formula for prime numbers. This is wrong."

"Although the distribution of prime numbers shows certain regularity, such as Ulam spirals, in fact they show a certain degree of randomness in this regularity. Therefore, if you want to find the general formula of prime numbers, it is a Something that is completely impossible, at least at the moment.”

"The main role of proving the Riemann Hypothesis for us is to help us achieve a better estimate of the distribution of prime numbers based on the prime number theorem."

"If the Riemann Hypothesis holds true, we can give a more accurate upper bound estimate of the difference between π(x) and x/ln(x)."

"This will make it easier for us to find which areas among the infinite natural numbers are very likely to have such a prime number."

"As for whether the Riemann Hypothesis can be used to crack the RSA encryption password, that is also nonsense, because the Riemann Hypothesis itself can be assumed to be true and used directly. This situation is relatively common in related fields. It’s like mathematicians have found thousands of propositions, which were discovered based on the establishment of Riemann’s hypothesis.”

"So if the Riemann Hypothesis could help us break RSA encryption, it would have been used long ago."

At this time, some students raised their hands and asked doubtfully: "Teacher, since we can assume that the Riemann Hypothesis is true, why can't we just simply acquiesce that it is true? In this case, those more than a thousand propositions, not Can it be directly applied to mathematical research?”

Hearing this question, Xiao Yi smiled and said, "I believe many people will have questions like yours, but this goes back to the fundamental purpose of studying mathematics."

"Although it is not something worth publicizing, after all, for mathematicians, mathematics is indeed like a game specially designed to please themselves, not for the purpose of benefiting all mankind."

"Although the result is important, the process is also more important. Just like when you play games and read novels, it is never for the moment when the game ends, but for the enjoyment in the process."

“So, our mathematics pays attention to absolute correctness, not the correctness of assumptions. The correctness of assumptions seems to reflect a feeling that we are helpless with this problem, as if we will never have a chance to solve this problem. "

"This obviously underestimates our talented mathematics community. We must always believe that no matter how difficult the problem is, we will eventually solve it."

Listening to Xiao Yi's words, the mathematics majors present felt a surge of emotion.

Although it is still unknown whether they will continue to study mathematics in the future, what Xiao Yi said today has already left a deep impression on them. Perhaps in the next many years , even if they change majors, there is a high probability that they will always remember what they said today.

Maybe they can still make mathematics the white moonlight in their hearts.

However, at this time, a student asked with a smile: "Teacher, are you going to prove the Riemann Hypothesis?"

The students present suddenly brightened up and looked at Xiao Yi with expectant expressions.

Xiao Yi rolled his eyes and said angrily: "I will prove it tomorrow."

The tone was obviously a joke. The guys looked convinced and asked excitedly: "Is it true?"

Xiao Yi was speechless and said, "Okay, don't guess here. The Riemann hypothesis is not that easy to prove. There are a lot of mathematicians who study this problem. It can be said that it is the most studied of all mathematical conjectures. However, no one has been able to solve it until now. This is why it is considered the most difficult of the seven millennium problems."

"If you want to prove it, forget it. I don't plan to study it for the time being."

Hearing Xiao Yi say this, the students present showed a regretful expression.

Even their respected Professor Xiao did not plan to study this problem. It can be seen that this problem is really difficult.

As for them, they only know that this problem is difficult for the time being, but they have no idea how difficult it is.

...

I don't know when the bell rang for the end of get out of class.

Of course, it was only the first class, and there was a second class next, so Xiao Yi sat on it, waiting for these students to come up to the stage to ask him questions.

While answering questions, he continued to think about another way to implement the analytical extension that he had just thought of in his mind.

By using elliptic curves to complete the extension of the domain, more information can be revealed in the process, instead of ignoring some possible information in the process like directly performing analytical extension.

Analytical extension is more about finding more properties of related functions directly on the new domain, while his new method is more about finding some information that ordinary methods cannot find in this process of change.

Of course, the most important thing is to be able to connect with the content in algebraic geometry.

Xiao Yi began to calculate this process in his mind.

First...

You can try to combine the Riemann curve with it.

The relationship between the Riemann curve and the elliptic curve is quite close, especially these two concepts are important research objects in algebraic geometry.

With this thought in mind, Xiao Yi began the derivation process. Although the derivation process happened in his mind, it did not affect the efficiency at all, and at the same time, it did not affect his answering the questions of the students in front of him.

It is particularly easy for him to do two things at the same time.

However, he soon found that using the Riemann curve for research was not a good idea.

But he was very flexible in thinking and soon thought of something else.

Modular form.

The relationship between modular form and elliptic curve is quite close, thanks to a conjecture, which is now a theorem, called Taniyama Shimura Theorem.

It was proved by Andrew Wiles, and Andrew Wiles successfully proved Fermat's Last Theorem.

As for Taniyama Shimura Theorem, it means that all elliptic curves on Q are modular.

This means that every elliptic equation can be expressed in modular form, that is, the two can be equated.

In other words, now he can combine the process of using elliptic curves to show analytical extension with modular form, and after that, there are suddenly more methods.

After all, modular form can be called the fifth method of calculation besides addition, subtraction, multiplication and division. The important point is that its scope of use is very wide and can be connected with many concepts.

In the past, Xiao Yi used modular form in many places.

Xiao Yi's eyes suddenly lit up. In a moment, a lot of ideas emerged in his mind, waiting for him to try.

However, at this moment, the bell for class rang again, waking him up from his thoughts.

Well, it's time for class, so let's go to class first. As for the things to think about, let's think about them later.

At least, he now has some ideas, which is the best.

Maybe this is the most important step towards proving the Riemann hypothesis?

Well...

You still said you didn't study the Riemann hypothesis!

Thinking of denying that he was studying the Riemann hypothesis just now, Xiao Yi smiled slightly.

Then he stood up and said, "Class!"

At the beginning of a new class, Xiao Yi continued to talk about prime number distribution. In the last class, he mainly talked to them about some history and theory of prime number distribution, and also expanded on the Riemann hypothesis. There was not much practical stuff.

Of course, it is not completely meaningless to popularize the Riemann hypothesis to them. Otherwise, why would the Goldbach conjecture and the hail conjecture be included in the elementary school textbooks?

The main purpose is to stimulate students' interest in mathematics.

Maybe it can inspire these students to think, "So many mathematicians have not solved this problem, so if I can solve it, how great it would be", and then start to study mathematics hard.

Although it is basically impossible for most of them to do such a thing, it is a sure win if they can study hard for a period of time.

…

Time passed quickly.

In the second class, Xiao Yi mainly told students some methods in prime number distribution and how to use these methods to solve related problems.

His teaching style is still very attractive to students. Combining various methods, he can make these students in front of him more interested in thinking.

Every time he gives some examples, he will use some methods that seem very showy to solve these problems, which also makes these students feel quite amazing. At the same time, they will also work hard to learn such showy methods. Although this method is showy, the knowledge hidden in it can inspire people. If you can understand the ideas, you will benefit a lot.

After all, it is Xiao Yi's unique way of thinking. If you can learn a little, it will be a kind of growth for these students.

In this way, a class is over.

"Okay, this class is here. In addition, next week is the midterm exam. I remember telling you at the beginning of the school that this is related to 50% of your usual grades, so review well."

Hearing Xiao Yi's words, the students present immediately wailed.

The students in Hua Luogeng's class were naturally going to take an exam, and Xiao Yi was setting the questions. Although they had just entered the school, they had heard that Xiao Yi's questions were very difficult.

As for those who came to take the class for free, they were wailing because it meant that they would not be able to take Xiao Yi's class next week.

However, looking at the looks of these students in front of him, Xiao Yi just smiled and then left the classroom. Anyway, he never brought textbooks when he came to class.

He went straight back to the office and took a look at the laboratory next door. Only Liang Qiushi was there. Of course, the main reason was that Liang Qiushi was the only doctoral student under him.