For the gossip-starved mathematical community, any research progress on the Riemann conjecture will be talked about for a while.

The same is true for the West Coast's math mecca, the Department of Mathematics at the University of Los Angeles Berkeley.

As soon as the preprint of Professor Lu's paper was hung up in Arxiv, news of the proven Quasi-Riemann conjecture was everywhere, and anyone who sat in a café could hear excitedly discussing this century-old problem.

"Have you heard? The quasi-Riemann conjecture has been proven! ”

"I've just heard about it, it's unbelievable, no one has been able to translate the right side of the critical zone to the left even a little bit for a century!"

"The last research in this direction directly led to the proof of the prime number theorem, and I don't know what Professor Lu will bring us this time."

"Hey, if only I could have half the IQ of Professor Lu."

"Wake up, stop dreaming!"

Not only the students in the mathematics department at Berkeley, but also the professors are talking about it.

Although analytic number theory is not a strong direction for Berkeley, and it is still more biased towards partial differential equations, for scholars like Tao Zhexuan, who are hardly proficient in any field, it is natural that they will not miss the big news in the field of mathematics.

In the office of the mathematics department at Berkeley, Tao Zhexuan, sitting at a desk full of scratch paper, with a ballpoint pen in his hand, was staring at the densely written paper on the table with a frown, because he was so focused that he didn't even hear a knock on the door behind him.

The knock continued for a while, and then the back door was pushed open, and a tall, thin Asian man came in with a box of pizza. Judging by his skillful movements, he should be the doctor in this office.

"Professor, the pizza you ordered has arrived."

Without looking up, Tao Zhexuan said casually with his back to him.

"Just put it aside for me."

"Okay, although I recommend you eat it while it's hot," the PhD student glanced at the scratch paper in front of his boss and asked curiously, "Well, I looked at your topic on Mathoverflow this morning...... Are you still working on that paper? ”

Tao Zhexuan: "Yes. ”

Swallowing his saliva, the doctoral student asked in an expectant tone.

"Is there a result?"

"I'll tell you if I have one."

"Okay." Listening to this cold voice, the doctoral student stuck out his tongue, made a helpless expression, and withdrew from the door.

Generally speaking, Professor Tao is relatively approachable, and he is one of the few professors at Berkeley who will mingle with students.

However, this is also an occasional one.

Generally speaking, when he is stumped by something, don't expect him to be very enthusiastic even when his wife comes......

"It's kind of interesting."

"The introduction of algebraic geometry can be seen in his thesis in the shadow of Grothendieck's work, but it is not particularly obvious."

"The key lies in the selection of algebraic curves on the complex plane...... How did he come up with it? ”

"And...... Can this approach really be called proof? ”

Leaning back on the office chair and staring at the dusty ceiling fan overhead, Tao Zhexuan scratched the back of his head a little nerve-wracked.

Rarely did he bother with something for so long.

The ballpoint pen lightly touched the paper and muttered for a long time before he spit out a sentence with a solemn face.

"It's controversial."

However, the tone of the sentence was not affirmative, and he was rarely as confident as he had been when it came to the question.

"Am I wrong......

Or is it really as he thinks, and the proof is problematic?

……

Speaking of which, when he received Vera's email last time, Lu Zhou can't remember it very clearly.

Maybe a year ago?

Maybe it's something a little bit further away.

Apparently, Vera, who was studying the Riemann conjecture, naturally came across the preprint he had put out on Arxiv, and must have taken the time to study it, and some doubts arose from it.

Many things are difficult to discuss clearly in the email, and the two agreed to have a video call at 8 p.m. Beijing time the next day.

So......

After three years, Lu Zhou once again met his most proud and complicated disciple.

Although separated by a network cable......

At 8 o'clock the next night, the video call was opened.

Brewing a cup of coffee and sitting in the study, Lu Zhou was originally ready to discuss academic issues, but the atmosphere was a little unexpectedly stiff.

The little girl, who seemed to be holding back a lot of words and was about to speak, seemed to not know how to speak at this moment, so she sat there nervously, staring straight at the computer screen without speaking.

And when Vera was looking at Lu Zhou in the video with a worried face, Lu Zhou, who was also looking at Vera, had a subtle embarrassment on her face at the moment.

The blonde hair was combed into delicate princess braids, and the long skirt with silk edges outlined the curves of the shoulders, even Lu Zhou, who couldn't tell the difference between the lipstick numbers, could see that this look was obviously not everyday makeup, it was obviously carefully dressed.

What should I say at this time?

Long time no see?

It seems a little cold.

Your new hairstyle is great?

I always feel that there will be unnecessary misunderstandings......

Coughing slightly, Lu Zhou tried his best to make a formal expression and spoke in a clear voice.

"Let's get straight to the point...... Let me see where your doubts are. ”

Greetings and politeness will make the atmosphere a little stiffer, everyone is engaged in mathematics, and going straight to the point may make the atmosphere a little more relaxed.

Sure enough, with these words, the atmosphere in the video call softened a little.

Vera, who was sitting on the other side of the computer, was also relieved, and then nodded like a chicken pecking rice, and whispered.

"Uh-huh, please, please wait a minute......"

As she spoke, the little girl got up and walked briskly to the side, laboriously dragging a blackboard from the edge of the classroom.

Seeing the line of calculations on the blackboard, Lu Zhou was slightly stunned, and a trace of surprise quickly flashed in his eyes, and then his expression gradually became serious.

Looking at Lu Zhou, who was lost in thought, Vera cleared her throat and continued in a low voice, "This is the algebraic geometry method you used to prove the quasi-Riemann conjecture, and I tried to generalize the method from the problem itself. ”

After a quick glance at the contents on the blackboard, Lu Zhou nodded approvingly and gave a positive evaluation.

"It's done, it's good."

is worthy of being the apprentice he taught.

It's not easy to sort these things out.

Vera smiled embarrassedly: "Although Molina ...... Ms. Abel asked me not to talk to you about these issues, but I still think ...... It's better to discuss it with you. ”

Communicating this idea with a genius is actually a risky thing to do, because the academic community only recognizes the results, not where the "idea" comes from.

Therefore, not to mention some seemingly interesting ideas, sometimes even the phased results that are enough to make it to the top of the journal will be suppressed by the authors themselves...... If he has ambitions for higher goals, and he does not belong to the kind of scholar who is indifferent to fame and fortune and does not care about fame.

After a pause, Vera changed her serious expression and continued.

"I have carefully studied the algebraic curve you introduced on a finite field when you prove that there is no non-zero triviality of the ζ function in the region where the value of the ε is infinitely small and Re(s)≥1-ε exists, and I get the ...... based on the property of ζ(s)=2Γ(1-s)(2π)s-1sin(πs/2)ζ(1-s)"

As she spoke, Vera turned to face the blackboard and wrote out a line of equations in the blank space left in the corners.

Looking at Vera's board book on the blackboard, Lu Zhou's brows furrowed slightly, but he soon relaxed.

He already had a general idea of Vera's train of thought.

On the basis of her own proof method, she gives a Plancherel formula on a class of single-connected power-zero Lie groups through the mapping method of cohomology groups...... In short, he made some improvements to his group construction method, and then applied it to the amplification of ε values.

The chalk wandering on the blackboard gradually stopped, and Vera smiled shyly when she turned around.

"It probably works this way, but I can't do it halfway through. I think if it were you, you would be able to find the reason......"

"So...... If only I could help you. ”

In the last sentence, Vera's voice was very small, so small that she was not sure whether Lu Zhou on the other side of the video had really heard it.

After staring at the equation on the blackboard for a while, Lu Zhou, who was sitting there motionless, suddenly spoke.

"Do you know why you can't find the cause?"

Slightly stunned, Vera asked subconsciously.

“…… Why? ”

With a regretful expression on his face, Lu Zhou told the truth.

"Because of the group construction method, this problem cannot be proven."

Vera: "......?" ”