Hearing the compliment, the corners of Vera's mouth curled with a touch of happiness.

For her, this is undoubtedly the best compliment in the world.

Standing next to Lu Zhou, she said softly.

"Your guess is correct, the Kakutani conjecture is a number theory problem, and at the same time a complex analysis problem......

As early as 1994, L. dus proved that the 3n+1 conjecture is equivalent to the function equation h(z³)=h(z^6)+{h(z²)+λh(λz²)+λ²h(λ²z²)}/3z (where λ=e^(2πi/3)) in the unit disk {z:|z|The analytic function solution in <1} is in the form h(z)=h0+h1z/(1−z). (h0 and h2 are complex constants)

On this basis, Schleicher et al. (1998) proved that any integer function h(z) makes g(z)=z/2+(1−cosπz)(z+1/2)/2+1/π(1/2−cosπz)sinπz+h(z)sin²πz satisfies: N⊂Φ(g).

Based on these two conclusions, Vera proves the existence of the integer function h(z) by constructing an ingenious transcendent integer function, so that for each branch D containing a positive integer of g(z) and Φ(g) in the above conclusions, there is z0∈D, so that {g^ok(z0)}∞/k=1 converges to 1.

From this, it is not difficult to deduce, and the Kadotani conjecture is established!

"Very good proof ......" With a relieved smile on his face, Lu Zhou said from the bottom of his heart, "I am surprised by the excellence. ”

From the summer of '16, it's now the end of '17.

He was happy to see that his students had grown.

I am also glad to see that the theory of "group construction" that I constructed for additive number theory problems did not stop at Goldbach's conjecture, not only that, but also inherited from my own students.

He now has some experience of what kind of emotion the "joy" depicted by the system is.

“…… Thanks to your pointers. Curling the corners of her mouth, Vera said modestly, her eyes full of gratitude.

Although the process was completed by her, the entire proof idea was provided by Lu Zhou.

From the phased results of her report meeting at the University of Berkeley at the beginning of the year, to the final proof of cooperation with Qin Yue and Hardy, all the work has been carried out around this line of thought.

All the way to the end, she knows better than anyone how important these obscure jobs are.

Lu Zhou smiled: "Don't be modest, all I provide is direction, and it's you who runs to the end." ”

After a pause, he continued: "...... I recommend that you submit this paper to the Annals of Mathematics, but the editorial board is on vacation these days, so you can hang the paper on Arxiv first...... Perhaps the editors of the Annals of Mathematics will give up the rest of your vacation for you. ”

Hanging the paper on Arxiv as a preprint can play a role in taking advantage of the pit, but even if it is not hung, it doesn't really matter, Kakutani conjecture was very popular in the 80s and 90s, but it is not a popular research direction now, and the chance of hitting the pit is almost zero.

The reason why he made this suggestion, of course, Lu Zhou also had a little selfishness in it.

According to the inference he made when solving Goldbach's conjecture, the principle of the system's decision on task completion is based on the time when the paper was published.

So as long as the paper is hung on Arxiv, the task is complete.

After listening to Lu Zhou's words, Vera nodded seriously.

"I know, I'll do it right away."

Lu Zhou smiled: "Then that's it, as for the equation on the blackboard, I'll erase it for you...... Thank you for the gift. ”

Hearing this compliment, Vera's face showed a happy smile.

After a while, she suddenly lowered her head, and her cheeks flushed imperceptibly.

"Professor."

Lu Zhou: "What's wrong?" ”

Courage gradually rose in her heart, and she raised her head and looked at Lu Zhou.

"I ...... Can I make a wish to you? ”

"I'm not Santa Claus, and I probably won't be able to put what you want in your stockings on your nightstand." Lu Zhou made a joke, paused for a moment, and continued, "However, as long as it's not something that violates principles, and it's something I can do, I'll definitely help you." ”

"I ......"

Vera had just opened her little mouth.

However, it was then that she suddenly realized that her intentions might be troublesome to him.

The words that almost came to my lips somehow couldn't be spoken.

"I ...... Want to study for your PhD. ”

Slightly stunned, Lu Zhou immediately smiled and said, "Welcome!" Rather, if you suddenly want to change your mentor, it will down. ”

Taking some comfort from this, Vera smiled a little weakly.

For her own incompetence, she sighed silently in her heart.

Although I am more courageous now than I was two years ago......

But, is it still not enough?

……

After a long day of crazy fun at the Institute for Advanced Study in Princeton, Lu Zhou returned home with small gifts from students and colleagues.

The gifts are not expensive, they are all small gifts of no more than 30 US dollars, but they are light in value and expensive in affection.

As for the pens he gave to his students, he collected from conference sponsors when he traveled around the world to attend academic conferences.

These gifts are also not expensive, but they are full of commemorative significance.

After putting away these gifts, Lu Zhou sat on the sofa by the fireplace, closed his eyes and silently recited the system, sinking his consciousness into the system space.

Walking up to the translucent holographic panel, two lines of transparent text appeared in front of him.

【A.“ A study on the even coherence states of finite Vishilbert space q-distortion harmonic oscillators". Student: Wei Wen. Student engagement: 25%. Type of Experience: Mathematics, Physics. 】

[B. Kakutani conjecture topic.] Students: Vera, Qin Yue, Hardy. Student engagement: 70%. Type of Experience: Mathematics. 】

Before he went home, Vera uploaded the paper that had already been sorted out to Arxiv, which met the conditions of the system's judgment. Therefore, the entry about the Kakutani conjecture was also listed on the task list.

Although I don't know how the system determines who is its student, I have to say that it is quite accurate.

Not only that, but it can even quantify the proportion of their participation in the overall work.

"Is this a multiple-choice question?"

Looking at the two options on the screen, Lu Zhou, with his arms folded around his chest, muttered to himself thoughtfully.

According to the description of the task by the system, he can choose a paper to submit the task.

The total amount of experience awarded is determined based on the academic value of the propositions contained in the topic, and then multiplied by the coefficient addition of student engagement to calculate the final experience value.

In fact, it can be seen from the design of the task that the system hopes that he will cultivate more students who can complete research work independently, so he is required to avoid direct participation in the project during the task.

In fact, Lu Zhou is also avoiding this.

However, it now seems that even though he tried his best to avoid directly helping Vera and the others, the system's contribution to him was still 30%.

Because at the very beginning, he laid out a clear research direction for them, as well as a framework for the whole research idea.

And it was precisely by following his train of thought that they completed the final proof.

Originally, Lu Zhou was still a little expecting that the system might ignore this, but now it seems that he is purely thinking too much.

"You really don't have any loopholes for me."

Looking at the translucent holographic panel, Lu Zhou shook his head helplessly.

Without saying anything more, he reached out and checked "B", and in the dialog box that popped up later, he clicked the button to confirm the submission.

Soon, a pale blue light swept across the holographic screen, and a line of large, dreamlike words popped up.

[Congratulations to the host, complete the task!] 】