Chapter 389 Proof of the Kakutani Conjecture

Hearing that compliment, Vera's mouth curled up in a happy arc.

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

Standing next to Lu Zhou, she said softly.

"Your guess is correct. Kakutani's conjecture is a problem of number theory, and it is also a problem of complex analysis..."

As early as 1994, L.Berg and G.Meinardus proved that the 3n+1 conjecture is equivalent to the function equation h(z)=h(z^6)+{h(z)+λh(λz)+λh(λz )}/3z (where λ=e^(2πi/3)) has the form of h(z)=h0+h1z/(1z) in the analytical function solution of the unit disk {z:|z|<1}. (h0 and h2 are complex constants)

On this basis, D. Schleicher and others proved in 1998 that any whole function h(z) makes g(z)=z/2+(1cosπz)(z+1/2 )/2+1/π(1/2cosπz)sinπz+h(z)sinπz satisfies: NΦ(g).

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

It is not difficult to deduce from this, the Kakutani conjecture is established!

"Excellent proof..." With a gratified smile on his face, Lu Zhou said from the bottom of his heart, "I am surprised by the excellence."

From the summer of 2016 to the end of 2017.

He is very happy to see that his students have grown up.

I am also very pleased to see that the "group structure method" theory I constructed for the problem of additive number theory did not stop at Goldbach's conjecture, not only that, but also passed on to my students.

He now has a little understanding of what kind of emotion the "joy" described by the system is.

"...Thanks for your guidance." Curving the corner of her mouth, Vera said modestly, her eyes full of gratitude.

Although the process was completed by her, the whole idea of ​​the proof was provided by Lu Zhou.

From her phased results at the UC Berkeley report at the beginning of the year to the final proof in cooperation with Qin Yue and Hardy, all the work is 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: "There is no need to be humble, all I provide is the direction, and you will be the one who runs to the finish line."

After a pause, he continued: "...About this paper, I suggest you submit it to the "Annual Journal of Mathematics", but the editorial department is on vacation these days, you can hang the paper on Arxiv first... Who knows, the editor of the Annals of Mathematics will give up the rest of your vacation for you."

Posting a paper on Arxiv as a preprint can serve as a trap, but it doesn’t matter if you don’t post it. Kadoya’s conjecture was very popular in the 1980s and 1990s, but it’s not a popular research direction now. The chance of hitting a crater is almost zero.

The reason why Lu Zhou made this suggestion, of course, has his own selfishness in it.

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

So as long as the paper is posted on Arxiv, the task can be regarded as completed.

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

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

Lu Zhou smiled: "That's it, as for the calculations on the blackboard, I'll wipe them off for you...Thank you for the gift."

Hearing this praise, Vera showed a happy smile on her face.

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

"professor."

Lu Zhou: "What's wrong?"

Gradually plucking up the courage in her heart, she raised her head and looked at Lu Zhou.

"Can I...can I make a wish to you?"

"I'm not Santa Claus, I may not be able to put what you want in the socks on your bedside table." Lu Zhou joked, paused for a moment, and continued, "However, as long as it is not against the principle, and I can do it, and I will definitely help you."

"I……"

Vera just opened her small mouth.

However, at this moment, she suddenly realized that her thoughts might cause trouble to him.

The words, which were almost on the lips, somehow fell silent.

"I... want to do your PhD."

Slightly stunned, Lu Zhou smiled and said, "Welcome! Rather, if you suddenly want to change your mentor, it will hit me hard."

Gaining some comfort from these words, Vera smiled weakly.

For her disappointment, she sighed silently in her heart.

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

But, is it still not enough?

...

After spending a whole day at the Institute for Advanced Study in Princeton, Lu Zhou brought small gifts from his students and colleagues, and returned home dispirited.

The gifts are not expensive, they are all small gifts of no more than 30 US dollars, but the lightness lies in the value, and the most important lies in the friendship.

As for what he gave to the students, they were the pens he collected from conference sponsors when he attended academic conferences all over the world.

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

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

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

【A. "Research on the coherent states of q-distorted harmonic oscillators in finite-dimensional Hilbert spaces". Student: Wei Wen. Student participation: 25%. Type of Experience: Mathematics, Physics. 】

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

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

Although I don't know how the system judges which people are my students, I have to say that it judges quite accurately.

Not only that, it can even numerically calculate the proportion of their participation in the entire work.

"Is this a multiple-choice question?"

Looking at the two options on the screen, Lu Zhou, who had his arms around his chest, mumbled to himself thoughtfully.

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

The total amount of reward experience is determined according to the academic value of the propositions contained in the topic, and then multiplied by the coefficient of student participation 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 masters who can complete research work independently, so he is required to try his best to avoid directly participating in the project during the task.

In fact, Lu Zhou is also avoiding doing so.

But now it seems that even though he tried his best to avoid directly helping Vera and the others, the system's judgment on his contribution still reached 30%.

Because at the very beginning, he formulated a clear research direction and a framework for the entire research idea for them.

And they followed his train of thought to complete the final proof.

Originally, Lu Zhou had hoped that the system might ignore this point, but now it seems that he is simply thinking too much.

"It really doesn't give me any loopholes."

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

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

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

[Congratulations to the host, the task is completed! 】