For the mathematics community, any research progress on the Riemann's conjecture would be talked about for a long time.
This was no exception for the mathematics department at the University of California, Berkeley.
As soon as Professor Lu's thesis was posted on arXiv, news about the proof of the quasi Riemann's conjecture spread everywhere. You could hear people excitedly discussing this century-old conjecture.
"Did you hear? The quasi Riemann's conjecture was proven! "
"I just heard about it. It's unbelievable. No one has been able to move the right boundary of the critical zone to the left even a little bit!"
"The last research in this area directly led to the proof of the prime number theorem. I wonder what Professor Lu will bring us this time."
"Sigh, if only I had half of Professor Lu's IQ."
"Wake up, stop dreaming!"
It wasn't just the Berkeley mathematics students who were talking about this matter, but the professors were also talking about this matter.
Even though analytic number theory wasn't a strong field at Berkeley, Tao Zhexuan was a scholar who was proficient in almost every field. He obviously wouldn't miss out on big news in the mathematics world.
In the office of the mathematics department at Berkeley, Tao Zhexuan sat in front of a desk full of draft papers. He spun a pen in his hand as he stared at the densely written thesis on his desk. He was so focused that he didn't even hear the knock on his door.
The knocking continued for a while before the back door was pushed open. A tall and thin Asian man walked in with a box of pizza in his hand. Judging from his movements, he was probably the PhD student in this office.
"Professor, your pizza is here."
Tao Zhexuan didn't look up as he spoke casually.
"Just put it aside for me."
"Okay, but I recommend you eat it while it's hot." The PhD student placed the pizza on the table and looked at the draft paper in front of the boss. He then asked curiously, "Um, I saw your topic on Mathoverflow this morning … Are you still researching that thesis?"
Tao Zhexuan: "Yes."
The PhD student gulped and spoke in a hopeful tone.
"Any results?"
"If there is, I will tell you."
"Alright." Hearing this cold voice, the PhD student stuck out his tongue, made a helpless expression, and left the room.
Generally speaking, Professor Tao was quite approachable, and he was one of the big shots at Berkeley who got along well with the students.
However, this was also a matter of different fields.
Generally speaking, when he was stumped by a problem, even if his wife came, he wouldn't be very enthusiastic …
"Interesting."
"He introduced algebraic geometry. I can see the shadow of Grothendieck's works in his thesis, but it's not particularly obvious."
"The key is the choice of algebraic curves on the complex plane … How did he come up with it?"
"And … can this method really be called proof?"
Tao Zhexuan leaned back in his office chair and stared at the dusty ceiling fan. He scratched the back of his head.
It was rare for him to be bothered by something for so long.
The ballpoint pen lightly tapped on the paper. After pondering for a while, he spat out a sentence with a dignified expression.
"It's controversial."
However, he didn't speak in a positive tone. He wasn't as confident as he used to be.
"Did I read it wrong …?"
Or is it really like I thought, there's a problem with the proof?
…
Speaking of which, Lu Zhou couldn't remember the last time he received an email from Vera.
Maybe a year ago?
Perhaps it was something even further back in time.
Obviously, Vera, who was researching Riemann's conjecture, saw the preprint on arXiv. She must have spent some time researching it and had some questions.
A lot of things were difficult to discuss in an email. The two agreed to have a video call at 8 pm the next day.
As a result …
After three years, Lu Zhou once again saw his proudest disciple, who also made him feel the most complicated emotions.
Even though they were separated by an internet cable …
At 8 pm the next day, they had a video call.
Lu Zhou made a cup of coffee and sat in his study room. He was prepared to discuss academic issues, but the atmosphere was a bit stiff.
The little girl who seemed to have a lot of things to say didn't know what to say at this moment. She just sat there nervously, staring at the computer screen without saying a word.
While Vera was looking at Lu Zhou in the video, Lu Zhou was also looking at Vera. He had an awkward expression on his face.
Her blonde hair was combed into an exquisite princess braid, and her long silk dress outlined the beautiful curves of her shoulders. Even though Lu Zhou couldn't tell the difference between lipstick and lipstick, he could tell that she obviously didn't wear makeup on her face. She obviously dressed up.
What should I say at this time?
Long time no see?
Seems a bit cold.
Your new hairstyle is pretty good?
He felt that there would be unnecessary misunderstandings …
Lu Zhou coughed and tried to make a formal expression. He spoke in a clear voice.
"Let's get straight to the point … Let me see what your doubts are."
Small talk would make the atmosphere a little stiff. They were all mathematicians, so if they went straight to the point, it might make the atmosphere a little more relaxed.
Sure enough, after he said that, the atmosphere in the video call softened a little.
Vera, who was sitting on the other side of the computer, also let out a sigh of relief. Then, she nodded like a chick pecking rice and whispered.
"Yes, yes, please wait a moment …"
As she spoke, the little girl got up and walked to the side with quick steps. She laboriously dragged a blackboard filled with words from the side of the classroom.
When Lu Zhou saw the lines of calculations on the blackboard, he was stunned for a second. His eyes flashed with a hint of surprise, and his expression gradually became serious.
Looking at Lu Zhou, who was deep in thought, Vera cleared her throat and said quietly, "This is the algebraic geometry method you used to prove the quasi Riemann conjecture. I tried to generalize the method from the problem itself."
Lu Zhou quickly scanned the contents on the blackboard and nodded with approval. He gave a positive evaluation.
"Well done."
No wonder she's my student.
It's not easy to organize these things.
Vera smiled embarrassedly and said, "Even though Molina … Ms. Abel told me not to discuss these problems with you, I still think … it's better to discuss it with you."
Discussing ideas with geniuses was actually a risky thing. This was because the academic community only recognized the results, not where the "idea" came from.
Therefore, not to mention some interesting ideas, sometimes even in-progress results that could be published in top journals would be suppressed by the authors themselves … If the authors were ambitious for a higher goal, and they weren't the kind of scholars who didn't care about fame and fortune and didn't care about their status.
Vera paused for a second and put on a serious expression.
"I carefully studied the algebraic curve on the finite field that you introduced when ε takes an infinitesimal value and there is no non-zero trivial point of the ζ function in the region where Re (s) ≥ 1-ε, and according to the property that ζ (s) = 2Γ (1-s) (2π) s-1sin (πs/2) ζ (1-s) …"
Vera turned around and faced the blackboard. She left a blank space in the corner and wrote lines of calculations.
Lu Zhou looked at Vera's writing on the blackboard and frowned, but he quickly relaxed.
He roughly understood Vera's idea.
Based on his own proof method, she gave the Plancherel formula on a class of simply connected nilpotent Lie groups through the mapping method of homology groups … Basically, she made some improvements to his Group Structure Method and used it to amplify the value of ε.
The chalk on the blackboard gradually stopped, and Vera turned around and smiled shyly.
"It's probably going to work this way, but I can only do half of it before I can't go on. I think if it was you, you would definitely be able to find the reason …
"So … it would be great if I could help you."
Vera's voice was so quiet that she wasn't sure if Lu Zhou really heard her.
Lu Zhou stared at the calculations on the blackboard for a while. He sat there motionlessly and suddenly spoke.
"Do you know why I can't find the reason?"
Vera paused for a second and asked.
"… Why?"
With a regretful expression on his face, Lu Zhou told the truth.
"Because the Group Structure Method can't prove this problem."
Vera: "…?"
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