Scholar's Advanced Technological System
chapter-845

845 Three Years! Lu Zhou stared at his mission panel for five minutes, and in the end, he decided to activate the mission card.

Even though the Lunar Orbit Committee planned on building a mass driver on the moon, he had no idea how long it would take.

He should be using this time to complete another mission instead.

After all, the lunar mass driver was advancing forward by itself, so the mission could be picked up again at any time.

[Golden reward mission: Activated!

[Description: The beginning of a future era starts with mathematics…

[Requirements: Solve the Riemann hypothesis within three years!

[Mission rewards: 10,000 general points, two million mathematics experience points. “Legendary” mission card.]

“… Solve the Riemann hypothesis in three years?”

Lu Zhou finished reading the translucent mission panel and muttered to himself, “I know this is the crown of mathematics, but three years…

“Is more than enough time.”

Lu Zhou double-checked the mission requirements again. He then tapped the screen and closed his mission panel.

Solving the Riemann hypothesis wasn’t an easy task. Even though he already solved the Quasi Riemann hypothesis, climbing the final part of the mountain would take a lot of effort.

But why was Lu Zhou so confident?

Because there had yet to be a problem that took him more than three years to solve…

Lu Zhou had no doubt that he could solve this problem within three years.

This was both his mathematics intuition and his self-confidence from being the king of modern mathematics!

“The ‘legendary’ mission card sounds exciting…”

Surely legendary is better than golden, right?

Lu Zhou didn’t know what was hidden behind that mission card, but the word legendary made him thrilled…

…

After Lu Zhou exited the system space, he opened his eyes and woke up in his office.

He felt a warm sensation climbing from his spine to his brain. It was like his neurons were immersed in a spa of knowledge. He never felt better before.

It felt like…

He was one step closer to becoming the omniscient God.

It didn’t take long for the information to enter his brain, and the warm sensation in his spine gradually subsided.

Lu Zhou moved his shoulders and felt something weighing on him. He reached out and felt a blanket.

He looked at the girl in the office. The girl blushed and said, “I saw you were sleeping, so I put the blanket on you.”

Lu Zhou looked at Han Mengqi and smiled.

“Thank you.”

“You’re welcome… Oh, the question you assigned me, I finished it.”

Han Mengqi was turning bright red. She tried to avoid eye contact as she walked up and handed him the stack of A4 papers.

“I don’t know if it’s right, but… I thought of it myself.”

“Let me see.”

Lu Zhou took the stack of A4 papers from the girl and glanced at it.

The title was the question he assigned her.

[For any real number s u003e 1, define ζ(s) = Σ1 / (m ^ s)… Prove that ζ(2n) is a transcendental number.]

Lu Zhou spent five minutes looking through the first couple of pages. He then gave her an evaluation.

“Standard proof.”

Lu Zhou looked at the calendar, then looked at Han Mengqi.

“I’m surprised. I thought it would take you more time to prove it, I didn’t expect you to finish it this year.”

Han Mengqi couldn’t help but smile proudly. She pouted and replied, “I’m actually pretty smart.”

Lu Zhou smiled.

“I agree with that.”

Lu Zhou looked like he had some questions, so Han Mengqi energetically spoke first.

“Go ahead, ask away!”

“Line 16, page three.”

Han Mengqi quickly found the line on her A4 copy.

Lu Zhou picked up the room temperature coffee cup on his table and took a sip. He paused for a second before saying, “Explain in detail on how you introduced the ζ (2n) from equation 2 as a transcendental number.”

After hearing this question, Han Mengqi was relieved.

She did a ton of preparation before coming to Lu Zhou, so she didn’t expect Lu Zhou to ask a fairly basic question.

She took a deep breath and replied, “This can be obtained by transforming equation 2 using Euler’s formula. For any integer nu003e 1, ζ (2n) = b (n) π ^ (2n).

“B(2n) is a sequence of rational numbers, which is, Bernoulli numbers. Obviously ζ (2) is π ^ 2 times a special rational number, and ζ (4) is π ^ 4 times a special rational number… So it is obvious that ζ (2), ζ (4)… are rational numbers. And because π is a transcendental number, the function values are also transcendental numbers.”

After hearing Han Mengqi’s explanation, Lu Zhou nodded with approval.

“Not bad.

“Don’t be happy just yet, that was just to prove you wrote the thesis yourself. The following question is the real challenge.”

Lu Zhou put down his coffee cup and spoke.

“Now that you have proven that ζ (2n) is a transcendental number, I want to ask, what about ζ (3)?”

What a simple question…

Han Mengqi proudly raised her chin.

However, when she was about to answer the question, she froze.

ζ (3)!

ζ (3)!

What what what?

What is that?

Han Mengqi was muddled, Lu Zhou smiled and asked, “Can’t answer it? ζ(3) seems simpler than ζ (2n), right? It doesn’t even contain a variable.”

“Yeah…” Han Mengqi pondered. She didn’t know what to say.

After a while, she spoke in an uncertain tone.

“Maybe… it’s also a transcendental number?”

Lu Zhou smiled and said, “Oh really? Why?”

Han Mengqi answered honestly, “It was a guess.”

Seeing the girl lower her head, Lu Zhou smiled and spoke.

“It’s not surprising you don’t know the answer. Because Euler also didn’t know. It wasn’t until 1978, when French mathematician R. Apery proved that ζ (3) is not a rational number. As for whether or not ζ (5) is a rational number, we still don’t know.”

After Han Mengqi heard that there was no answer to the question, she pouted.

“What is that… There isn’t even an answer to the question… You’re bullying me.”

“There is an answer.” Lu Zhou smiled at Han Mengqi and said in a serious manner, “There’s an answer to every mathematics problem, we just don’t know it. When you become a PhD student, that is where the challenge is. You will have to find your own ideas for proofs, then find the proofs themselves.”

Han Mengqi paused for a second.

She immediately realized what was going on, and she looked ecstatic.

“Wait a second, you’re saying that I can be your student!”

Lu Zhou smiled and nodded.

“I actually already made up my mind after you answered my first question.

“The second question will be your research project.”

Lu Zhou stood up from his desk and walked to the blackboard. He picked up a piece of chalk and wrote on the blackboard as he spoke.

“The transcendence value of the Riemann zeta function at odd positive integers has always been a classic problem in analytical mathematical theory. According to Euler’s formula and the properties of Bernoulli numbers, we can easily prove that ζ (2n) is a transcendental number. Therefore, our hypothesis is that for any integer n u003e 1, ζ (2n + 1) is also a transcendental number.

“The best result so far is that there are countless ζ (2n + 1), which are irrational numbers. However, the difference between infinities is still infinity.

“If you can do good research in this area, even if it’s only a small proof, you will be recognized by the academic community.

“By then, you will be able to graduate.”