The top student must be diligent.

Chapter 104 Witness History
Just as the mathematicians below are very clear about this, as the author of this paper, Xiao Yi is even more aware of how important Part 1 is in his entire paper.

Therefore, he also meticulously broke down even every small step of this part and narrated it in detail.

As time goes by, the content becomes more and more critical.

His voice was the only voice left in the entire venue, while everyone else seemed to be attending a classical music concert, not even daring to cough.

"... Now we start to consider twisted sums over arithmetic series, where we can obtain results where the coefficient b is replaced by any complex number αrs, provided that the range of the modulus D is neither too small nor too large."

"This means that, for category D, we can treat the sums S1 and S2 simultaneously, since the Jacobi symbol (s/r) can be incorporated into βz."

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

“So we need some new ideas here.”

"Automorphic theory of etale algebraic varieties."

At this point, Xiao Yi paused and made a joke: "In the past period of time, a group called [Beneficiaries of the Theory of Automorphism of Etale Algebraic Clusters] has emerged in the mathematics community, and now it seems that I have become one of them."

There was laughter from the audience.

However, the laughter basically came from those people in the back who only half understood what was being said, or who had no idea at all.

As for the big guys sitting in the front row, they just managed to curl their lips, thinking in their hearts: Don’t talk about it, the next step is the most critical one, hurry up and tell me!

Xiao Yi did not disappoint them, and continued: "In the next step, we need to express its automorphic form, then introduce the trace formula and extract the Etale fundamental group algebraic variety..."

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

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

"..."

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

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

"I see. I have always been confused. According to the Barban-Davenport-Halberstam theorem, if the old congruence modulus d is in the range of (RS)^1/2- to RS(log RS)2A, then the new modulus m is in the range processed in the first step. In that case, it is obviously impossible to match the range of category D. So this is how he considered this step!"

"Etale algebraic variety automorphism theory can also be used like this!"

An old mathematician with a head full of white hair said with amazement to another 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 the GPY sieve, which was used to prove that there are infinite sets of prime numbers whose intervals are arbitrarily smaller than the average interval of prime numbers. Later, Zhang Yitang also achieved a breakthrough based on the GPY sieve.

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

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

"In this way, using duality and the Poisson summation method... what a brilliant move!"

Goldstone nodded: "Next, we just need to construct two new polynomials... and the classification work is done!"

For a moment, Goldstone's eyes even turned red.
Ivanets noticed this but didn't say anything.

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

The two old mathematicians studied the sieve method for their entire lives, but were also stumped by the damn parity check problem all their lives.

They all knew that it would be basically impossible to prove the twin prime conjecture by the sieve method without finding a way to reduce the impact of the parity check problem to a sufficiently low level.

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

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

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

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

He once thought that he would never see the parity check problem being suppressed to a small enough level in his lifetime. He also expressed his regret more than once in interviews: "I think I will never see the day when the twin prime conjecture is proved in my lifetime."

And now, he thought, he saw it.
……

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

There were also other analytic number theorists, or experts in sieve theory methods, whose expressions were more or less excited about Xiao Yi's move.

If they hadn't been afraid of making any noise and disturbing Xiao Yi's report on the stage, they might have applauded.

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

Somewhere in the second row, Terence Tao's eyes were shining and he was almost jumping up and dancing.

Then he turned his head to look to both sides and asked, "What do you think?" Sitting next to him were Zhang Yitang and Maynard. As collaborators who had once worked together to push Zhang Yitang's result to 246, they also sat together at this twin prime number report meeting.

Maynard was also very excited. He nodded repeatedly and said, "Indeed! Hey, Tao, don't you remember the problems we encountered in our project? If we use this method, the problem we are facing now seems to have a chance to be solved!"

"That's exactly what I wanted to say!" Tao Terence nodded repeatedly.

Zhang Yitang, who was standing next to him, listened to the excited conversation between the two and couldn't help but smile and shook his head. Tao was twenty years younger than him, and Maynard was thirty years younger, so they were relatively more energetic than him.

Although he was excited about Xiao Yi's move, he did not show as much enthusiasm as they did.

Of course, he also got some inspiration from Xiao Yi's step. If he used such mathematical thinking, some of the difficulties he faced in the Landau-Siegel zero point conjecture might be able to get further breakthroughs.

"Perhaps, I should bring Xiao Yi with me to study this problem together?"

This idea came to Zhang Yitang's mind, and he immediately felt that it was feasible.

With Xiao Yi's inspiration that always pops up from time to time and his strong technical ability, it will definitely be of great help in solving the Landau-Siegel zero-point conjecture.

Generally speaking, independent researchers like Zhang Yitang, who have already made many interim results in their research projects, are often not willing to invite others to work with them. After all, once the results are out, it means sharing the honor together.

However, at this moment, Zhang Yitang felt that if he invited Xiao Yi to participate in his project, the interim results he had achieved in the past would probably soon become insignificant.

So it's totally worth it.

However, just as he had just said this thought in his mind, he heard Terence Tao next to him say: "Hey, James, do you think we should invite Xiao Yi to join us? I believe he will be of great help to us."

Maynard agreed as soon as he heard it: "Of course, this is definitely a good idea! In fact, I wanted to tell you this, but I was just afraid that you would disagree."

"Of course I agree to this!"

"..."

Hearing the conversation between the two, Zhang Yitang was stunned. This time Xiao Yi might become popular.

It seems that after today's report meeting is over, he will go to Xiao Yi to discuss this matter.

……

Time passed quickly, and the most important part of the report was finished, and the following content became smoother.

Of course, Xiao Yi will still try his best to explain those important steps clearly enough.

And so it went on until the end.

"…Finally, we can use the most typical estimation method to complete the estimation of formula 14.15, and obtain the formula 14.17 and 14.18…"

【lim(n→∞) inf(pn+1pn)<3】

After writing down the formula that appeared at the beginning of the report, Xiao Yi put down the pen in his hand, turned around neatly, and faced all the audience present with a smile.

"In 1919, Norwegian mathematician Vigo Brown proved that the sum of the reciprocals of all twin primes converges to a mathematical constant, known as Brown's constant, which is only approximately equal to 1.902. This is undoubtedly a very small number, which is enough to let us know how rare the distribution of twin prime pairs is."

"But finally, until today, we have reason to believe that among the infinite prime numbers, there will always be prime number pairs with a difference of only 2!"

"One of the eighth questions Hilbert asked at the International Congress of Mathematicians 120 years ago. I think that's the end of it!"

Xiao Yi opened his arms, as if making an announcement to the entire mathematical community.

"Of course, at the end, my friends, if you have any questions about my proof, you can ask them now."

He looked at the more than 1000 spectators in front of him.

However, as time passed, no one raised their hand to ask questions.
For those who don’t understand, they don’t know what questions to ask.

And for those mathematicians who understood, their questions have been answered in the report!
It was not until a moment later that Faltings, who was sitting in the first row, stood up first and applauded, just like he did at the Max Planck Institute for Mathematics.

The other people also woke up as if from a dream, stood up and gave the warmest applause.

For the genius who is bowing gracefully on the stage.

But also for this elegant and stunning conclusion!

They truly witnessed history!

……

【Please give me a monthly ticket! 】

(End of this chapter)