841 Debatable?

Any research progress regarding Riemann’s hypothesis would be talked about in the mathematics community.

This was no exception for the mathematics department at UC Berkeley.

After Professor Lu uploaded his thesis onto arXis, the news regarding the proof of the Quasi Riemann hypothesis spread like wildfire. People were discussing the news all over campus.

“Did you hear? The Quasi Riemann hypothesis was proven!”

“I did, that’s incredible. No one has made any progress for a century!”

“Last time the prime number theorem was proved as a result of research on the Riemann hypothesis. I wonder what Professor Lu will make of it this time.”

“Yeah, if only I had Professor Lu’s intelligence.”

“Keep dreaming!”

The mathematics department students weren’t the only ones talking about this event. The professors were talking about it as well.

Analytic number theory wasn’t Berkeley’s strong suit. They leaned toward the partial differential equations side. However, for scholars like Tao Zhexuan, who was proficient in multiple fields, they obviously kept up with the latest news in mathematics.

Tao Zhexuan sat inside an office in the mathematics building at Berkeley. His desk was full of draft papers. With a pen in his hand, he stared intently at the thesis on his table. He was so focused he didn’t even hear the knock on his door.

After a while, the door was pushed open. A thin Asian man walked in with a box of pizza. He was a PhD student.

“Professor, the pizza you ordered is here.”

Tao Zhexuan replied without looking up, “Just put it aside.”

“Okay, but you should eat it while it’s hot.” The PhD student looked at the thesis in front of him and asked curiously, “Oh yeah, yesterday, I saw your comment on Mathoverflow… Are you still reading that thesis?”

Tao Zhexuan: “Yes.”

The PhD student gulped and asked, “Is there a conclusion yet?”

“I would tell you if there were one.”

“Okay then.” The PhD student gave a helpless expression then left the room.

Normally speaking, Professor Tao was an approachable person. He was one of the big names at Berkeley who were friends with the students.

However, there were exceptions.

Whenever he was troubled by a question, he would act cold toward everyone, even his wife…

“Interesting.

“He’s using an algebraic geometry, similar to something Grothendieck would do, but it’s not obvious.

“Choosing an algebraic curve on the complex plane… How did he come up with this idea?

“Also… Can he really prove it using this method?”

Tao Zhexuan leaned back on his chair and looked at the dusty ceiling fan. He scratched his head.

It was rare for a problem to bother him for this long.

He tapped the pen on the paper and solemnly spoke.

“Debatable.”

However, he didn’t sound confident at all.

“Am I the one who’s wrong…”

Maybe the proof does in fact, exist?

…

Speaking of which, Lu Zhou didn’t remember when was the last time he received an email from Vera.

Was it a year ago?

Or even longer?

Obviously, Vera, who was researching Riemann’s hypothesis, saw Lu Zhou’s thesis on arXiv. She took the time to read it and had some questions.

Most things were difficult to convey through email. Therefore, the two agreed to a video call at 8 pm Beijing time.

After three years, Lu Zhou could finally see the student he was proudest of.

Even though it was through the Internet…

At 8 pm the next day, the video call began.

Lu Zhou made a cup of coffee and sat in his study room. He was prepared to discuss academic problems, but the atmosphere was a bit stiff.

It seemed like the girl had too much to say, but she didn’t know where to start. She nervously sat in front of the computer screen without saying anything.

Lu Zhou had an awkward look on his face.

Her blonde hair was combed into a ponytail. Lu Zhou could even tell she was wearing a different shade of lipstick. Clearly, she dressed up.

What should I say?

Long time no see?

That seems a bit cold.

Nice hairstyle?

What if she thinks I’m hitting on her?

Lu Zhou coughed and spoke clearly in a formal way.

“Let’s get to it… Tell me what your question is.”

Having small talks and greetings might make the atmosphere even stiffer. Therefore, he went straight to the topic.

As expected, the tension in the room disappeared.

Vera sighed in relief and nodded. She spoke quietly.

“Okay, wait a second…”

The girl walked to the side and dragged a fully-written blackboard into the camera view.

Lu Zhou looked at the lines of equations and paused for a second. He looked surprised, then gradually, he started to look serious.

Vera cleared her throat and said quietly, “This is the algebraic geometry method you used to prove the Quasi Riemann hypothesis, I tried to summarize the proof.”

Lu Zhou skimmed through the blackboard and nodded with approval.

“Not bad.”

No wonder she’s my star student.

Being able to summarize all this is not easy.

Vera smiled and said, “Even though Molina… I mean Ms. Abel told me not to discuss these problems with you, I think it’s better… to discuss it with you.”

Talking about ideas with a genius scholar was risky, because the academic community only recognized who produced the result, not where the initial idea came from.

Therefore, some scholars even resisted the urge to publish their in-progress results… They wanted to aim for higher, something more significant.

Vera put on a serious face and spoke.

“I carefully studied the algebraic finite field curve you introduced when ε tends to infinitely small. I found out that there are no non-zero trivial points of the ζ function in the region of Re (s) ≥ 1-ε. The properties of 2Γ(1-s) = (2π)(s-1)sin (πs / 2) ζ (1-s) hold…”

Vera turned toward the blackboard and wrote on the right corner, which she purposely left blank.

Lu Zhou looked at Vera and frowned, but he quickly relaxed.

He had a rough idea of Vera’s thought process.

She used his own proof as a foundation and introduced a homology mapping function that connected the nilpotent Lie groups… Basically, she made some improvements on his Group Structure Method and applied that to the enlargement of ε.

The chalk in Vera’s hand stopped moving. She turned around and smiled.

“This method will probably work, but I can only do it halfway. I was thinking if you could finish it…

“So… I need your help.”

Vera spoke quietly. She wasn’t even sure if Lu Zhou could hear her.

Lu Zhou stared at the formula on the blackboard for a while. He suddenly asked, “Do you know why you can’t complete the proof?”

Vera paused for a second and asked, “Why?”

Lu Zhou spoke the truth with a look of regret.

“Because you can’t prove this problem with the Group Structure Method.”

Vera: “…?”

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