The top student must be diligent

Young genius, can he succeed?

As the applause rang out like a tide, people once again asked in their hearts.

Xiao Yi looked at the more than a thousand people in the audience and took a deep breath for a moment.

There are really a lot of people...

Last time, there were only more than 600 people at his lecture at the Institute of Mathematics, but this time it was doubled.

It is said that there are only a little over a thousand people, but in fact, there are probably at least 1,200 people.

However, although there are many spectators, it did not bring him much pressure, nor did it make him nervous.

In his opinion, these people are here to listen to his knowledge sharing, so why should he feel pressure?

Walking to the center of the podium, he bowed slightly and then raised his head.

Smiled: "Ladies and gentlemen, thank you for coming to this report meeting, I am Xiao Yi."

"Since the beginning of last month, my article "Parity Check Sorting Sieve and Twin Prime Conjecture" has attracted the attention of the mathematical community, so this time Princeton University also prepared this report meeting for me, which is my honor."

"To express my gratitude, I will do my best to make this report satisfy everyone."

"Then, the report begins."

"First, state our ultimate goal, the twin prime conjecture."

Turning around and walking to the big blackboard behind him, Xiao Yi wrote a formula on it.

[lim(n→∞) inf(pn+1pn)\u003c3】

"That is to say, there are infinite pairs of prime numbers with a difference less than 3."

"And of course, considering the properties of natural numbers and prime numbers, it is equivalent to: there are infinite pairs of prime numbers with a difference of 2."

"This is also a description of the twin prime conjecture."

"The study of prime numbers has been going on for many years since ancient times. More than 2,000 years ago, Euclid proved that there are infinite prime numbers. Later, there were problems such as the Mersenne prime conjecture and the Goldbach conjecture."

"Until later, mathematicians saw that there were always prime pairs with a difference of 2 appearing continuously, so they began to think, are there infinite pairs of such prime numbers?"

"In 1900, Hilbert formally proposed this problem, and until now, many mathematicians have tried to solve it."

"Now, 120 years have passed, and I think it's time to put an end to it."

The PPT that had been prepared long ago turned a page at this time.

[Sieve method and parity check problem]

"To solve the problem of twin prime conjecture, we need to use a tool, that is, the sieve method."

"I will not go into details about what the sieve method is, but mainly explain what the parity check problem in the sieve method is."

"..."

Xiao Yi slowly narrated, and gradually introduced the audience to the main topic of the report.

After explaining what the parity check problem is, he briefly introduced how mathematicians in the past tried to avoid this problem.

Finally, Xiao Yi said: "However, if we always try to avoid this problem, although it can alleviate its impact, it will ultimately prevent us from making greater progress."

"This is probably the key to our stagnation in the study of the twin prime conjecture."

"So, it's time to face it."

"Since the sieve method in the past made it impossible for us to distinguish between natural numbers with odd prime factors and even prime factors, then why not think about how to classify such situations more finely?"

The mathematicians in the audience who had faced this problem shook their heads helplessly.

Of course they had considered it, but how to achieve it was a huge technical difficulty, and it lasted until more than two months ago.

If it weren't for the few steps that Xiao Yi simply revealed in the video, they would probably still be kept in the dark.

At this time, Xiao Yi turned around and wrote a few formulas on the blackboard.

[∑aplogp=HA(x){1+O(log(x)/log(x)……]

These mathematicians immediately narrowed their eyes and stopped distracting.

Because they all knew that the key point was coming.

If you want to ask where the most critical part of Xiao Yi's paper is, then it can only be the part that proves the classification of natural numbers with odd and even prime factors at the beginning.

The key point that the parity check classification sieve can be established is also in this part!

Can Xiao Yi's report answer the questions in their hearts?

Chapter 104 Witnessing history

Just as the mathematicians below are very clear, as the author of this paper, Xiao Yi is more aware of how important Part 1 is in his entire paper.

Therefore, he also meticulously split this part, even every small step, and explained it carefully.

As time passed, the content became more and more critical.

Only his voice was left in the audience, and the others seemed to be attending a classical music concert, and they didn't even dare to cough.

"... Now we begin to consider the twisted sum of arithmetic series. When the modulus category D is neither too small nor too large, we can obtain the result of replacing the coefficient b with any complex number αrs."

"This means that, for category D, we can process the sums S1 and S2 simultaneously because the Jacobi symbols (s/r) can be merged into βz."

"If αrs are factored, this will be a Barban-type result and will follow the large sieve given earlier."

"So here we need some new ideas."

"The automorphic theory of etale algebraic varieties."

Having said this, Xiao Yi paused and made a joke: "In the past period of time, a group called [Beneficiaries of etale algebraic variety automorphic theory] appeared in the mathematics world, and now looking at it, I myself have become one of them. a member.”

There was a burst of laughter in the audience.

However, the laughter basically comes from those people in the back who half understand or are completely ignorant.

As for the big guys sitting in the front row, they reluctantly raised their lips and wanted to say: Don't pull away, the next step is the most critical step, hurry up!

And Xiao Yi did not disappoint them, and then continued: "The new step is to express its automorphic form, and then introduce the trace formula and extract the etale basic group algebraic variety..."

[∑_(qQ)_(q,a)=1~γqB(q)Ax/(log x)^A...]

"Then, considering duality, Poisson summation, and some basic but non-trivial independent variables, we get a new category (log RS)^A"

"..."

"Combining these results, we successfully find every bound for S1 and S2 that, contrary to the trivial bound, will save an amount (log RS)^A over the entire range for any A!"

As Xiao Yi's narration came here, the eyes of the mathematicians sitting in the front row, especially the top experts in analytic number theory, all lit up.

"That's it. I have always been confused before. According to the Barban-Davenport-Halberstam theorem, if the old congruential modulus d is in the range of (RS)^1/2- to RS(log RS)2A, then the new modulus The quantity m is within the scope of the first step. In that case, it obviously cannot match the scope of category D. It turns out that this is what he considered in this step!"

"The automorphic theory of etale algebraic varieties can still be used in this way!"

An old mathematician with gray hair looked amazed and said to another similarly old mathematician next to him.

This old mathematician, named Dan Goldstone, is also an expert in sieve theory. More than ten years ago, he and two other mathematicians jointly proposed a method called GPY sieve, which was used to prove that It is deduced that there are infinite groups of prime numbers, and their intervals are arbitrarily smaller than the average interval of prime numbers. Moreover, Zhang Yitang later achieved a breakthrough based on the GPY sieve.

As for the old mathematician next to him, it was Henryk Ivanets, and the two of them were considered old friends.

Hearing Goldstone's words, Ivanets nodded in agreement and said, "Yes, I remember we discussed this issue before. What I didn't expect was that he could actually think of introducing some non-trivial independent variables! "

"In this way, borrowing the method of duality and Poisson summation... what a wonderful ingenuity!"

Goldstone nodded: "Next, as long as we construct two new polynomials... the classification work is complete!"

For a moment, Goldstone's eyes even turned red.

Ivanets noticed this, but did not say it.

Because he felt the same way, he didn't even know if his eyes were red.

Two old mathematicians have studied the sieve method all their lives, and have been stumped by the damn parity problem their whole lives.

They all know that it is basically impossible to prove the twin prime conjecture through the sieve method without finding a way to reduce the impact of the parity problem to a sufficiently low level.

This is why Zhang Yitang's original method could only stay at the number 246 after continuous optimization.

As for the idea of ​​proving the twin prime conjecture without using the sieve method...

Let’s not talk about whether there is such a possibility. Even if it is, at least it has not appeared in the current mathematical world. Maybe only God can give the answer - of course, it does not rule out the possibility that even God cannot give the answer. sex.

Goldstone looked at the young figure on the stage with an expression of gratitude.

He once thought that he would never see the parity problem suppressed to a small enough level in his life. He also expressed his regret more than once in interviews: "I think I will never see twins in my life." The day the prime number conjecture was proven.”

And now, he thought, he saw it.

…

Goldstone and Ivanets were not the only ones who were excited.

There were other analytical number theorists or experts in sieve theory methods, and their expressions at this time were more or less excited as Xiao Yi made this step.

If it weren't for the fact that the noise could not affect Xiao Yi's report on the stage, maybe they would have applauded.

"As expected of the founder of the automorphic theory of etale algebraic clusters, his step of introducing the category D into the automorphic representation in the form of trace formulas and then extracting the basic group information... Wow! This technology is probably only found and implemented by him now?"

Somewhere in the second row, Terence Tao's eyes lit up and he almost jumped up and danced.