Scholar's Advanced Technological System

Chapter 830: On-Site Question

Teaching undergraduate students was like a knowledge review session for Lu Zhou.

Normally, he wouldn’t touch on these rudimentary materials. This was the only time he set aside his research and focused on the more simple concepts.

“… We all know that the Riemann hypothesis is one of the most important conjectures in the analytic number theory. It is a conjecture about the zero points of the zeta function. But, do any of you know how the Riemann hypothesis came to be?

“In fact, prior to the Riemann hypothesis, there was another proposition that troubled mathematicians for centuries. That is, the distribution of prime numbers.”

Lu Zhou wrote down a few equations on the board. He then looked back at the students in the classroom and continued, “Using the fundamental theorem of arithmetic, even high school students know that every positive integer can be expressed as the product of prime factors. This representation is unique, thus prime numbers are the basic elements that make up a positive integer.

“However, the distribution of prime numbers is not as easy to understand. One of the most basic tasks of the analytic number theory world is to study the distribution of prime numbers.”

The students had a look of concentration on their faces, Lu Zhou knew his lecture was going well.

The Riemann hypothesis was indeed a complex problem, understanding it was difficult, and solving it was near impossible…

Lu Zhou paused for a second. He then continued, “In analytic number theory, mathematicians often study the function π(x), a function that outputs the number of primes lower than x. Researching the characteristics of π(x) is one of the central tasks of the analytic number theory.

“Both Gauss and Legendre have done a lot of numerical calculations on π(x). They guessed that when x tends to infinity, π(x)~x/ln(x). Their conjecture was later proved, which is what we now understand as the Prime Number Theorem.

“Euclid proved that there are infinitely many prime numbers. Euler introduced the Euler product. These great pioneers provided us the tools to analyze and study prime numbers. No one was able to find a suitable method for proving Gauss’ conjecture. That was until the 1950s, when a German mathematician published a paper titled ‘On the exact number of primes less than a given limit’. His research opened a new road for π(x).

“Most people know who this German guy is, that’s right, I’m talking about Riemann. He introduced the Riemann zeta function in this thesis.”

Lu Zhou turned around and wrote down an equation on the blackboard.

[ζ(s)=Σ1/n^s]

Lu Zhou looked at the dead silent classroom and spoke.

“This is it… It doesn’t look hard at all, right?”

Everyone: “…”

F*ck sake!

How is it not hard?

“Riemann made a further hypothesis for the function he proposed, which was that all of the zero points of ζ(s) are on the critical straight line. It turns out his vision was quite revolutionary. All of our brute force calculations show that the zeros points are on a critical straight line. Unfortunately, even though we know that his hypothesis is likely correct, we have no way to prove it.

“We can often use the Riemann hypothesis to prove other conjectures. However, if the Riemann hypothesis isn’t proven, we can’t say for certain that other conjectures are correct.

“Vice versa, if we do prove the Riemann hypothesis, then thousands of mathematical conjectures that assume the Riemann hypothesis will become theorems!

“If anyone can prove the Riemann hypothesis, they would undoubtedly become the greatest mathematician of this century… I am certain of that, even though this century has just begun.”

“Professor,” a student said with his hand raised in the air. After receiving a nod from Lu Zhou, he asked excitedly, “If someone solves the Riemann hypothesis, how will they compare to you?”

“It won’t be a good comparison. After all, my work goes beyond just the field of mathematics.” Lu Zhou smiled at the student and said, “But if anyone does prove the conjecture, their work in mathematics will undoubtedly surpass mine.”

After that, Lu Zhou explained some of the current research progress on the Riemann hypothesis. Because he changed his lecture style, the students were listening more intently.

Lu Zhou was satisfied with the performance of his students.

The time quickly passed by.

Lu Zhou glanced at the clock on the wall and saw that it was almost time to end his class. He chucked his chalk onto his desk and spoke.

“We’ll end it here… Class dismissed.”

Shuffling sounds of textbooks filled the classroom. Lu Zhou nodded toward the students, grabbed his lesson plan, and walked out of the classroom.

Lu Zhou was about to go back to his office. He wanted to write down his inspirations, which he got from the lecture. However, Dean Qin suddenly appeared out of nowhere.

“Excellent lecture!” Dean Qin said with a smile on his face. “It helped me a lot!”

Lu Zhou smiled.

“You’re too kind, I haven’t taught undergraduate students in a while.”

Dean Qin said, “We all have our own priorities, and your research is obviously more important than lecturing. Speaking of which, are you busy these days?”

Lu Zhou: “Not really, why?”

“I have something to ask you.” Dean Qin coughed and said, “Have you heard of the International Mathematical Olympiad?”

Lu Zhou: “I have, why?”

He had obviously heard of IMO. Unfortunately, he didn’t have the chance to attend.

The IMO gold medalists were the best of the best.

For example, Schultz, who Faltings said was one of the three people who could surpass Faltings himself, was an IMO gold medalist.

As for why Schultz signed up for two more IMO tournaments after winning a gold medal… It was because Schultz thought it was fun…

Dean Qin smiled and said, “Here’s the thing, last month there was a national high school mathematics competition, right? The top students in every state have been selected. The winter training camp will start in January next year. It’s already November, so it’s time to weed some people out.”

Lu Zhou said, “You’re not asking me to write exam questions, right?”

Dean Qin: “This wasn’t my decision, the China Mathematics Society wants you to come up with the final problem.”

Lu Zhou: “Is that appropriate?”

Dean Qin smiled and said, “Of course it’s appropriate. The final question last year was also chosen by an academician. Not only are you an academician, but you’re also a Fields medalist.”

Lu Zhou: “Fine, it’s only one question anyway.”

“Yeah, thanks.” Dean Qin suddenly remembered something and said, “Oh yeah, don’t make it too difficult. There’s no point if no one can solve it.”

“Don’t worry, I won’t make it too difficult.” Lu Zhou pulled out a piece of draft paper from his lesson plan and began writing.

Dean Qin looked at him, perplexed.

“You’re not going to write the question right now, right?”

Lu Zhou: “Of course I am, why?”

Dean Qin said, “This is the national final competition, so you should think about it carefully.”

“I just did.” Lu Zhou wrote down the question and handed it to Dean Qin. “Give this to China Mathematics Society. It should be fine.”

Dean Qin stared at the piece of draft paper. Lu Zhou began to walk away as Dean Qin muttered to himself, “Riemann zeta function?”

Dean Qin rubbed his chin and thought to himself.

“Can high school students even solve this problem?”

However, he suddenly realized something, and his eyes lit up as he spoke.

“Wait a second… This question is interesting…”

Dean Qin carefully looked around him and stuffed the paper into his pocket. He then quickly walked back to his office.