The top student must be diligent.

Chapter 274 The Road to Proving the Riemann Hypothesis

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

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

Then, Xiao Yi began to write on the blackboard.

By combining the Riemann zeta function with the prime number theorem, the form of the Riemann hypothesis is finally revealed, and then it is explained how much the Riemann subhypothesis is related to the distribution of prime numbers.

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

"Although the distribution of prime numbers shows a certain regularity, such as the Ulam spiral, they actually show a certain randomness within this regularity. Therefore, it is completely impossible to find a general formula for prime numbers, at least not at the moment."

"The most important thing about proving the Riemann hypothesis is that it can help us achieve a better estimate of the distribution of prime numbers based on the prime number theorem."

"If the Riemann hypothesis holds, we can give a more precise upper bound on the difference between π(x) and x/ln(x)."

"This will make it easier for us to find which regions in the infinite natural numbers are very likely to contain such a prime number."

"As for whether the Riemann hypothesis can be used to crack RSA encryption, that is also nonsense, because we can assume that the Riemann hypothesis itself is true and use it directly. This is a common thing in related fields. For example, mathematicians have found thousands of propositions that are discovered based on the premise of the Riemann hypothesis."

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

At this time, a student raised his hand and asked in confusion: "Teacher, since we can assume that the Riemann hypothesis is true, why can't we just assume that it is true? In this way, wouldn't the more than one thousand propositions be directly applied to mathematical research?"

Hearing this question, Xiao Yi smiled and said, "I believe many people will have the same question as you, but this involves the fundamental purpose of our study of mathematics."

“Although it is not something worth promoting, after all, mathematics is like a game for mathematicians to please themselves, not for the benefit of all mankind.”

“The result is important, but the process is even more important. Just like when you play games or read novels, you never play for the moment of the game ending, but for the enjoyment of the process.”

"So, our mathematics emphasizes absolute correctness, not hypothetical correctness. Hypothetical correctness seems to reflect a feeling that we are helpless with this problem, as if we will never have the chance to solve it."

"This obviously underestimates our mathematical community, which is full of talented people. We must always believe that no matter how difficult the problem is, we will eventually be able to solve it."

Listening to Xiao Yi's words, the mathematics students 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 many years to come, even if they change their majors, they will probably remember what he said today.

Maybe mathematics can always be the white moonlight in their hearts.

However, at this time, a student asked with a smile: "Teacher, when will you prove the Riemann hypothesis?"

The eyes of the students present suddenly lit up, and they all looked at Xiao Yi with expectant expressions.

Xiao Yi rolled his eyes and said unhappily, "Proof it tomorrow."

It was obviously a joke, but the guys looked like they believed it and asked excitedly, "Is this true?"

Xiao Yi was speechless and said, "Okay, stop guessing here. The Riemann hypothesis is not that easy to prove. There are a lot of mathematicians who have studied 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 have no plans to study it for the time being anyway."

Hearing what Xiao Yi said, the students present showed expressions of regret.

Even their respected Professor Xiao has no intention of studying this problem. It can be seen that this problem is really difficult.

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

……

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

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

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

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 as in the direct analytical extension.

Analytic extension is more about discovering more properties of related functions directly on the new domain of definition, while his new method is more about finding out some information that cannot be discovered by ordinary methods 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 first try to combine it with the Riemann curve.

The relationship between Riemann curves and elliptic curves is quite close, especially since both concepts are important research objects in algebraic geometry.

With this thought in mind, Xiao Yi began the deduction process. Although the deduction process took place in his mind, it did not affect the efficiency at all. At the same time, it did not affect his answers to the questions from the students in front of him.

For him now, doing two things at the same time is extremely easy.

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

But he had a very flexible mind and soon thought of something else.

Model form.

The relationship between modular forms and elliptic curves is quite close, thanks to a conjecture, which is now a theorem called the Taniyama-Shimura Theorem.

It was proved by Andrew Wiles and led to his successful proof of Fermat's Last Theorem.

As for Taniyama-Shimura's 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, he can now combine the process of using elliptic curves to show analytical extension with modular forms, and after that, there are suddenly more methods. After all, modular forms can be called the fifth method of calculation besides addition, subtraction, multiplication and division. The important point is that it has a very wide range of applications and can be connected with many concepts.

In the past, Xiao Yi made use of the modeling form in many places.

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

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

Well, it’s time to attend class, so just focus on the class first. As for the things that need to be thought about, let’s leave them for later.

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

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

Ok……

You also said that you didn't study the Riemann hypothesis!

Thinking of the fact that he had just denied that he was studying the Riemann hypothesis, Xiao Yi smiled slightly.

Then he stood up and said, "Class!"

At the beginning of a new class, Xiao Yi continued to talk about the distribution of prime numbers. 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. Not much practical stuff was talked about.

Of course, it is not completely meaningless to educate them about questions like the Riemann hypothesis. Otherwise, why would we include questions like the Goldbach conjecture and the hail conjecture in elementary school textbooks?

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

Maybe this will inspire these students to think, “So many mathematicians haven’t solved this problem, so it would be so great if I could solve it,” and then they will 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 it can allow them to study well 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 more interested when thinking.

Every time he gave some example questions, he would use some seemingly fancy methods to solve them, which amazed the students. At the same time, they would also work hard to learn such fancy methods. Although these methods are fancy, the knowledge hidden in them is very inspiring. If you can understand the ideas behind them, you will benefit a lot.

After all, that was Xiao Yi's unique way of thinking. If these students could learn a little bit from it, it would be a kind of growth.

And just like that, the class ended.

"Okay, that's it for this class. Also, next week is the midterm exam. I remember telling you at the beginning of the semester that this exam affects 50% of your regular grades, so study hard."

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

The students in Hua Luogeng's class were naturally taking an exam, and the questions were set by Xiao Yi. Although they had just entered the school, they had heard that the questions Xiao Yi set were extremely difficult.

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

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

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

However, this year's admission to graduate school work has already been completed. It is estimated that there will be an exchange activity between the admitted students and their supervisors, and he will probably continue to recruit a few new graduate students at that time.

He will probably recruit new graduate students at this time every year from now on, unless there is another major project like the Japan-Creating Project.

Finally, he sat down at his desk and skillfully took out the draft paper. Then, he summarized the method he had just thought of in class.

At the same time, because he is not in class now, he has more time to think about the overall process of this method and continue to improve it further.

Soon, he perfected it.

"It should be OK now."

What should this new method be called?

Xiao Yi thought about it for a while, and finally decided to simply name it elliptical anticurve analysis.

Because after extending its domain, if we redraw the ellipse, it can no longer be called an ellipse, but rather a combination of an ellipse and an inverse curve.

This is mainly because its domain exceeds the major axis of the ellipse itself.

"Well, now we can combine the modular form with this new elliptical form..."

Xiao Yi continued to deduce.

And sure enough, just as he had expected, once the two were combined, a broader mathematical vision was revealed - at least that was Xiao Yi's mathematical vision.

This means that he will be able to process this information in more ways.

After just thinking for a moment, Xiao Yi immediately thought of something, and this thing was what he had just used some time ago.

That is the Geometric Langlands Program!

By using the proven geometric Langlands conjecture, perhaps he could combine these methods to create a more powerful tool?

A long-lost wave of emotion suddenly arose in Xiao Yi's heart.

Because his intuition told him that this would give him the opportunity to lead him to the path of truly proving the Riemann hypothesis!
……

(End of this chapter)