For Lu Zhou, having a class with an undergraduate student was also a review of his knowledge.
In the past, these were obvious things to him, and he basically wouldn't even consider them. It was only at this time that he could temporarily put aside his research and think about why the obvious things were obvious.
"… Many people know that Riemann's conjecture is one of the most important and difficult conjectures in analytic number theory. It is a hypothesis about the distribution of zeros of the Riemann zeta function. However, very few people know why Riemann's conjecture was proposed.
"In fact, before Riemann's conjecture, there was a larger proposition that had been studied by countless scholars for centuries, which was the distribution law of prime numbers."
Lu Zhou wrote a few numbers on the blackboard and looked back at the students in the classroom.
"Through the most basic arithmetic theorem, even middle school students know that every positive integer can be expressed as a product of prime factors. If you don't consider the order of the prime factors, then this representation is unique. Therefore, prime numbers have become the basic elements of positive integers.
"However, the distribution law 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 field of analytic number theory is to study the distribution law of prime numbers. "
Lu Zhou looked at the students in the classroom and knew that his class was almost half successful.
Although Riemann's conjecture was a very complicated problem, it was not as difficult as most people thought. The real difficulty was how to solve it …
Lu Zhou paused for a second and continued.
"In analytic number theory, people usually study the function π (x) and use it to represent the number of prime numbers that do not exceed x. Studying the distribution law of prime numbers is the basic task of analytic number theory, and studying the behavior of π (x) is the central problem of analytic number theory.
"Regarding the problem of π (x), Gauss and Legendre both did a lot of numerical calculations and conjectured that when x tends to infinity, π (x) ~ x/lnx. This conjecture was later proven, and it is now known as the prime number theorem."
"Euclid used an elementary method to prove that there are infinitely many prime numbers, and Euler introduced a product formula. These pioneers provided the possibility of analyzing and studying the prime number problem. However, until the 1850s, people did not find a suitable method to prove the conjecture proposed by Gauss. It was not until a German mathematician published a paper titled 'On the number of prime numbers not exceeding a given value' that a new path was opened for the study of π (x).
"Many people have already guessed who this big name is. Yes, he is the Riemann I'm talking about. The Riemann zeta function that he introduced in this paper influenced the next century and a half."
Lu Zhou turned to the blackboard and wrote down a line of calculations.
[ζ (s) = Σ1/n ^ s]
Lu Zhou looked around the silent classroom and spoke.
"This thing … It doesn't look that difficult, right?"
Students: "…"
MMP！
How is it not difficult?!
"In his paper, Riemann made a further conjecture on the function he proposed. He believed that all of the non-obvious zeros of ζ (s) are on the critical line. It turns out that his vision is quite far-sighted. All of the non-obvious zeros obtained after a lot of calculations are on the critical line. Unfortunately, although we know that there is a high probability that it is correct, there is no way to prove that it is correct.
"Therefore, we can often get a very beautiful result under Riemann's conjecture. However, if we can't prove that Riemann's conjecture is true, we can't prove that our results are correct.
"The opposite is also true. If we can prove that Riemann's conjecture is correct, then thousands of mathematical conjectures based on Riemann's conjecture will be promoted to theorems!
"If anyone can prove Riemann's conjecture, they will undoubtedly become the greatest mathematician of this century … Even though this century has just begun."
"Professor," a student couldn't help but raise his hand. After receiving a nod from Lu Zhou, he stood up and asked excitedly, "What if someone can prove Riemann's conjecture, compared to you?"
"It's not easy to compare. After all, my achievements are not only in the field of mathematics," said Lu Zhou as he looked at the student who asked the question. He smiled and said, "But if someone can really prove this conjecture, then their achievements in the field of mathematics will undoubtedly stand at the top of this era."
Then, Lu Zhou continued to talk about some research progress on Riemann's conjecture, as well as some of its equivalents. Even though it was all boring stuff, perhaps because of the change in teaching methods, the students were clearly listening more attentively than last class.
Lu Zhou nodded proudly in his heart. He was satisfied that he had gradually regained some of his form.
Time quickly passed by, and soon, it was time for the class to end.
Lu Zhou looked at the clock on the wall and saw that it was almost time. He threw the chalk in his hand on the multimedia desk.
"Let's end the class here. Just now, I happened to have some new ideas … So, class dismissed."
The sound of textbooks being put away could be heard in the classroom. Lu Zhou put his lesson plan under his elbow and nodded at the students who were watching him leave. He then turned around and walked out of the classroom.
Lu Zhou walked out of the classroom and was about to go back to his office. He wanted to write down the inspiration he had in class. Suddenly, Dean Qin appeared out of nowhere and stopped him.
"Excellent number theory class!" Dean Qin walked up to Lu Zhou with a smile on his face and spoke emotionally. "Even I benefited a lot from listening to it."
Lu Zhou smiled awkwardly.
"You're too kind. I'm ashamed to say that I haven't taught undergraduates in a while. "
Dean Qin said, "Everything has a priority. This is related to the national interest. Compared to teaching the students, your research is more important. Speaking of which, are you busy? "
Lu Zhou: "I'm not very busy, why?"
"Here's the thing, I have something to ask of you." Dean Qin coughed and said, "Have you heard of the Olympic Mathematics Competition?"
Lu Zhou: "I've heard of it, is there a problem?"
Of course he had heard of the IMO competition, even though he had never participated.
Every year, the people who won the IMO gold medal were godlike athletes.
For example, Schulz, who Faltings said was one of the three most promising young scholars in the world, had won three IMO gold medals in a row.
When people asked him why he signed up for two more IMO competitions after winning the gold medal, he said it was because it was fun …
Dean Qin smiled and said, "Here's the thing, there was a national high school mathematics league last month, right? The top few students from each province had already been selected. By the national finals of the winter camp in January next year, about 30 students would be selected to enter the national training team. It's already November, it's time to prepare the exam questions. "
Lu Zhou paused for a second before he smiled and said, "You're not asking me to be the question question maker, right?"
Dean Qin: "I'm not in charge of this, it's an invitation from the China Mathematics Society. They want you to be the question maker for the last question."
Lu Zhou: "Do you think it's appropriate for me to be the question maker?"
Dean Qin smiled and said, "What's wrong with that? In the past, the final question was also set by an academician. Not only are you an academician, but you're also a Fields Medal winner. If you can't do it, who else can?"
Lu Zhou: "Okay then, if it's just the last question."
"Okay, I'll leave this to you." Dean Qin suddenly remembered something and quickly added, "Oh yeah, don't make it too difficult. If no one can do it, then it's pointless."
"Don't worry, it won't be too difficult," said Lu Zhou as he took out a piece of draft paper from the lesson plan under his elbow. He tore off a corner and began to write on it.
Dean Qin looked at Lu Zhou and paused for a second. He didn't know whether to laugh or cry.
"You're not planning on setting the question here, right?"
Lu Zhou: "What else?"
Dean Qin said, "This is the national final, I have to think about it carefully."
"I've thought about it." Lu Zhou wrote the question on the paper and folded it. He then gently stuffed it into Dean Qin's hand and said, "Give it to the China Mathematics Society, this question should be fine."
Dean Qin stared at Lu Zhou as he walked away. He then looked at the draft paper in his hand and quietly opened it.
"Riemann zeta function?"
Dean Qin rubbed his chin and muttered to himself.
"Can a competition student solve this kind of question?"
However, at this moment, he seemed to have thought of something, and his eyes lit up slightly.
"Oh … Wait a second, this question is interesting …"
Dean Qin carefully folded the paper and stuffed it into his pocket. He then turned around and walked toward his office.
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