Chapter 831

For Lu Zhou, taking classes with undergraduates is also a kind of review of knowledge points.

As usual, these things are obvious to him, and basically he would not think about them. Only at this time can he temporarily put aside the research itself and think about why the obvious things are so obvious.

"...Many people know that the Riemann conjecture is one of the most important and difficult conjectures in analytic number theory. It is a hypothesis about the zero point distribution of the Riemann zeta function. But few people know that the Riemann conjecture is why And was brought up?"

"In fact, before the Riemann Hypothesis, there was a larger proposition that has been studied by countless scholars for centuries, that is, the distribution law of prime numbers."

As he spoke, Lu Zhou wrote down a few numbers on the blackboard, then looked back at the students in the classroom and continued.

"Through the most basic arithmetic theorem, even junior high school students know that every positive integer can be expressed as the product of prime factors. If the order of prime factors is not considered, then this representation is unique, so prime numbers have also become The basic elements that make up positive integers."

"However, the distribution of prime numbers is not as easy to understand as its definition. It can even be said that one of the most basic tasks of the entire discipline of analytic number theory is to study the distribution of prime numbers."

Seeing the students in the classroom gradually enter the state, Lu Zhou knew that his class was almost half successful.

Although the Riemann Hypothesis is a very complicated problem, it is not as difficult as most people imagine to understand it. The real difficulty is how to solve it...

After a pause, Lu Zhou continued.

"In analytic number theory, people usually study the function π(x), and use it to represent the number of prime numbers not exceeding x. Studying the distribution of prime numbers is the basic task of analytic number theory, and studying the properties of π(x) , is the central problem of analytic number theory.”

"On the issue of π(x), both Gauss and Legendre have done a lot of numerical calculations, and guessed that when x tends to infinity, π(x)~x/lnx, this conjecture was later proved, which is what we understand the prime number theorem."

"Euclidean used elementary methods to prove that there are infinitely many prime numbers, and Euler introduced a product formula. These pioneers provided the possibility for the analysis and study of prime numbers. However, until the 1850s, people did not Find a suitable method to prove the conjecture proposed by Gauss until a German mathematician published a paper entitled "On the Number of Prime Numbers Not Exceeding a Given Value", which is the research on π(x) opened up a new path.”

"Many people may have already guessed who this big cow is. Yes, he is the Riemann I want to talk about, and the Riemann zeta function he introduced in this paper has influenced the next century and a half. "

Said that Lu Zhou turned to face the blackboard, and wrote a line of calculations on the blackboard.

【ζ(s)=Σ1/n^s】

Looking around the silent classroom, Lu Zhou continued.

"That's the thing...doesn't look too hard, does it?"

Students: "..."

MMP!

Where is it not difficult? !

"Riemann made a further conjecture on the function he proposed in the paper, and believed that all the non-obvious zeros of ζ(s) were on the critical straight line. Facts have proved that his vision is indeed quite far-sighted. After a large number of calculations, the obtained All the non-obvious zeros are on the critical line. Unfortunately, although we know that it has a high probability of being correct, we have no way to prove that it is indeed correct."

"So, we can often get a very nice result under the Riemann Hypothesis, but if we can't prove the Riemann Hypothesis is true, we can't prove that our result is correct."

"The reverse is also true. If we can prove that Riemann's conjecture is correct, then thousands of mathematical conjectures that exist assuming Riemann's conjecture will be promoted to theorems!"

"If anyone can prove the Riemann Hypothesis, he will undoubtedly become the greatest mathematician of this century... I can say this responsibly, even though this century has just begun."

"Professor," at this time, a student couldn't help but raised his hand. After getting a nod from Lu Zhou, he stood up and asked excitedly, "If you can prove the Riemann conjecture, how does it compare with you?"

"This is not easy to compare. After all, my achievements are not only in the field of mathematics," Lu Zhou smiled at the student who asked the question, and gave an uncertain statement, "But if someone can really prove this conjecture If so, then his achievements in the field of mathematics will undoubtedly stand at the pinnacle of this era."

Next, Lu Zhou continued to talk about some research progress on the Riemann Hypothesis, as well as some of its equivalent forms. Although they are all boring things, perhaps because of the change of the teaching method, the students obviously listened more seriously than the previous class.

Satisfied that he has gradually recovered some state, Lu Zhou nodded proudly in his heart.

Time passed by every minute and every second, and soon it was time to finish class.

Lu Zhou glanced at the wall clock on the wall and saw that the time was almost up, so he threw the chalk in his hand on the multimedia desk.

"Let's end this class here. I happened to have some new ideas just now... Then, get out of class is over."

The sound of sparsely packing up the textbooks sounded in the classroom. Lu Zhou clamped the lesson plans under his elbows, nodded to the leaving students who watched him leave, then turned and walked out of the classroom.

Walking outside the classroom, Lu Zhou was about to go back to the office to record the inspiration he had just had in class when he was suddenly stopped by Dean Qin who came out of nowhere.

"A very good number theory class!" Coming forward with a smile on his face, Dean Qin said with emotion, "Even if I finish listening to it, I will benefit a lot."

Lu Zhou smiled embarrassedly and said.

"Thank you. I'm ashamed to say that I haven't taught undergraduates in a while."

Dean Qin said understandingly: "Everything has its own priority and is related to national interests. Compared with teaching students, your research is of course more important. Speaking of which, have you been busy recently?"

Lu Zhou: "Is it very busy, what's the matter?"

"That's right, there is one thing I would like to ask you for," Dean Qin coughed lightly, and said, "Have you heard of the Olympic mathematics competition?"

Lu Zhou: "I've heard of it, but I've heard of it. Do you have any questions?"

Of course he has heard of the IMO competition, although he has never had the chance to participate.

Those who win the gold medal in the IMO competition every year are all players who are like gods.

For example, the Schulz he knew, who was called by Faltings as one of the three young scholars most likely to surpass him in the world, had won three consecutive IMO gold medals.

As for being asked why he signed up for two sessions after winning the gold medal, according to him, it was because he thought it was fun...

Dean Qin smiled and said, "That's right. Didn't we hold the National High School Mathematics League last month? The top few in each province have already been selected. By the national finals of the winter camp in January next year, about 30 people will be selected." Students enter the national training team. Now it’s already November, and it’s almost time to prepare for the exam questions.”

Lu Zhou froze for a moment, then laughed and said, "...you asked me to be the one to make a proposition, didn't you?"

Dean Qin: "I don't care about this matter. It's an invitation from the Huaguo Mathematics Society. They hope that you will be the proposer for the last final question."

Lu Zhou: "Is it appropriate for me to be the proposition person?"

Dean Qin said with a smile: "Is this suitable or not? In previous years, the last question in the national finals was also proposed by an academician. You are not only an academician, but also a Fields Medal winner. If you are not suitable, then there is Who can be suitable?"

Lu Zhou: "That's fine, it's just the last question."

"Well, I'll leave this to you." Suddenly remembering something, Dean Qin hurriedly added, "By the way, don't make it too difficult, it's meaningless if no one can do it gone."

"Don't worry, I won't make it too difficult," Lu Zhou said, pulling out a piece of draft paper from the lesson plan under his elbow, tearing off a corner and writing on it.

Seeing Lu Zhou take out the draft paper and start writing, Dean Qin froze for a moment, then couldn't tell whether to laugh or cry.

"You're not going to ask questions here, are you?"

Lu Zhou: "Otherwise?"

Dean Qin sweated: "This is the national final, so you have to think carefully about it."

"I've already thought about it," Lu Zhou wrote the topic on the paper, folded the draft paper, and gently stuffed it into Dean Qin's hands, "Pass it to the Huaguo Mathematical Society for me, there should be no problem with this topic. question."

Staring dumbfounded at the back of Lu Zhou leaving, Dean Qin shifted his gaze to the draft paper in his hand, opened it silently and looked at it.

"...Riemann zeta function?"

Mumbling something in his mouth, Dean Qin touched his chin, thinking to himself.

"Can contestants solve this kind of problem?"

However, at this moment, as if he had thought of something, his eyes lit up slightly.

"Hiss... wait, this question seems interesting..."

Looking around mysteriously, Dean Qin carefully folded the note and stuffed it into his pocket, then turned around and walked quickly towards the office.