The top student must be diligent

"Otherwise, I would have proved this problem long ago."

Faltings smiled slightly: "But, I have a suggestion that might be good."

"Why not think about it from the perspective of the Elliott-Halberstam conjecture?"

Chapter 73 Surrounded by four Fields Medal winners

"Elliott-Halberstam conjecture!"

Hearing Faltings' words, Xiao Yi was stunned.

The Elliott-Halberstam conjecture is also a conjecture about prime numbers in number theory.

And it is also of great significance to the mathematical community-it can help mathematicians save more effort when studying the distribution of prime numbers.

And the distribution of prime numbers directly refers to the most profound and influential problem in mathematics-the Riemann hypothesis.

So, proving the Elliott-Halberstam conjecture is very important to the mathematical community.

In the words of Terence Tao, proving this conjecture is a dream.

But obviously, dreams mean that it is difficult to achieve.

The purpose of the Elliott-Halberstam conjecture is to prove that the distribution level of prime numbers, θ, is less than 1. In the 1960s, Bombieri and another mathematician proved that this θ value is less than one-half, and there is still a gap of one-half from [1]. However, this gap of one-half seems to have become a chasm, which has deeply stumped the mathematical community and has never been able to make a breakthrough.

However, in modern times, this gap has gradually changed. For example, Zhang Yitang has made a breakthrough to 0.5017, thus completing the first great progress in the twin prime conjecture in one fell swoop.

Back to the twin prime conjecture, the reason why Faltings mentioned this conjecture to Xiao Yi is that once this conjecture is proved, the gap of the twin prime conjecture can be directly narrowed to 6.

To some extent, this result is equivalent to proving another conjecture, called the [Sexy Prime Conjecture]. A prime number pair with a difference of 2 is called a twin prime number, and a prime number pair with a difference of 6 is called a sexy prime number. Whether there are infinite prime number pairs with a difference of 6 is called the sexy prime number conjecture.

Although I don’t know why such prime number pairs are called sexy, maybe this is the aesthetic of mathematicians.

"Yes, it is the Elliott-Haberstam conjecture. Don’t you think your idea is particularly suitable for studying this problem?"

"Of course, I am not asking you to prove this conjecture, but just want you to think about whether it can help your current problem from the perspective of this conjecture."

"Far Abelian geometry, plus automorphic forms, I believe that these two things combined can play an unexpected role."

"What do you think?"

Faltings said with a smile, then picked up the cup next to him, drank a sip of water, and left time for Xiao Yi.

At this moment, Xiao Yi has also begun to think.

Faltings' suggestion suddenly inspired him.

This also made him exclaim that the old mathematician's experience was indeed quite powerful.

Although people praised his independent mathematical thinking and his lack of empiricism, experience can still play a huge role at certain times.

With the burst of inspiration, he began to write on the blackboard next to him a moment later.

[∑nxθ(n)λ(nhi)2=∑d1|P∑d2|Pμ……]

[θ(n)= Q2([d1，d2])φ([d1，d2])……]

After a while, the blackboard, which had little blank space left, was about to be used up by Xiao Yi.

However, at this moment, Faltings picked up the blackboard brush and helped him wipe off the previous handwriting.

At this time, Faltings was once again shocked by Xiao Yi's agile thinking.

He never expected that he had just said an idea, and the young man immediately had a thought in his mind.

Was it because of the young man's quick reaction or his extraordinary talent?

In the end, he could only sigh in his heart that he was getting old, and then continued to help Xiao Yi wipe the blackboard.

He didn't want to affect Xiao Yi's thinking because there was no place to write on the blackboard.

But at this moment, the door of the office was knocked, and Schultz walked in, followed by two people, Deligne and Bombieri.

"Professor, Professor Deligne and Professor Bombieri want to find you... eh?"

Seeing the scene in front of him, he was stunned.

Faltings, actually helping Xiao Yi wipe the blackboard?

Good guy, is Xiao Yi so favored?

He has never been treated like this!

And Deligne and Bombieri, who followed Schultz, were also surprised.

"Shh!"

Faltings saw the three people coming in, and immediately put his index finger in front of his mouth, asking them to be quiet and not disturb Xiao Yi.

At this time, he had also wiped the blackboard, leaving Xiao Yi with a whole blank again.

The three people who came in stopped talking, walked up and looked at it.

Soon, as an expert in this field, Bombieri was the first to figure out what Xiao Yi had written, followed by Deligne and Schulz.

Several people showed surprised expressions.

Bombieri: "He wants to study the Elliott-Halberstam conjecture from the perspective of far Abelian geometry?"

Deligne: "He actually knows how to keep his form?"

Schultz: "He actually did research on the Elliott-Halberstam conjecture?"

"Quiet, wait until he finishes writing." Faltings rolled his eyes and asked them to keep their voices down again.

Fortunately, Xiao Yi seemed to be completely immersed in the world of mathematics and didn't even notice anyone coming in.

The three Fields Medal winners finally calmed down and watched quietly. However, the more they watched, the more incredible they became.

Xiao Yi had unknowingly substituted far-Abelian geometry into the automorphic form of number theory, and began to combine the two.

Automorphic forms are an important concept in number theory, and applying automorphic forms to the study of prime number distribution is also a very important method in mathematics.

Mathematicians in China have previously won the second prize of the National Natural Science Award in 2014 for their research in this area, which has also brought very important inspiration to the mathematics community.

And now, Xiao Yi has undoubtedly achieved an important breakthrough in this direction!

Perhaps, based on such a breakthrough, the automorphic form can play a more important role in the distribution of prime numbers and even more and more extensive problems!

Finally, when the blackboard was about to be finished again, the pen in Xiao Yi's hand stopped.

"Well... we can only stop here for the time being."

After taking another look at the entire process on the blackboard, Xiao Yi touched his chin and finally put down the blackboard pen in his hand.

Just when he was about to speak to Faltings, he heard applause behind him.

"Incredible!"

"marvelous!"

"Perfect!"

Xiao Yi was stunned for a moment.

He turned around and saw three uninvited guests entering the room at some unknown time.

Four people gathered around him.

To elaborate more, four Fields Medal winners surrounded him as a minor boy.

"Uh, professors..."

But before he could say hello, Bombieri hurriedly asked: "What is your next idea?"

Chapter 74 Fa Shu: If Xiao Yi goes, I will go

Bombieri himself is an expert in the distribution of prime numbers.

As mentioned before, the θ of the prime number distribution level is less than one-half, which was discovered by him in the last century, so he has always been very concerned about research in this area.

And what Xiao Yi has shown now undoubtedly has a great chance to achieve a breakthrough in this.

After hearing Bangbieli's question, Xiao Yi gave up the idea of ​​saying hello and thought about the question for a moment in his mind, and then he answered.

"Using the relationship between far Abelian geometry and etale homology, we can introduce the Weil theorem proved by Professor Deligne. The next step is to further move from the ellipse to the sieve method. To solve this problem, we must inevitably It is the sieve method, which has always had a very critical effect on the problem of prime number distribution. "

"This is the next direction I can think of for the time being."

"The specific details may need to wait until I further complete the project before I can give more in-depth ideas."

Finally, Xiao Yi said slightly apologetically.

"That's enough, that's enough."

Bombieri murmured while looking at the derivation processes on the blackboard.

Until finally, he calmed down and said solemnly to Xiao Yi: "Your ideas have surpassed most people in this field, at least, just on the idea of ​​combining far Abelian geometry and automorphic forms , you have achieved an unprecedented innovation.”

"Perhaps you can really realize the dream of our number theory community and increase the value of θ of the Elliott-Halberstam conjecture to the most extreme level."

Next to them, three other Fields Medal winners, after hearing Bombieri's words, couldn't help but nodded and expressed their approval.

Indeed, they had never seen anyone achieve this level of success on this problem before, let alone combine it with far Abelian geometry.