Chapter 214 A Small Step to the Top of the Mountain

The full name of the circle method is "Hardy Littlewood's circle method". It is not only an important tool for studying Goldbach's conjecture, but also an important tool often used in analytical number theory.

And the invention of this tool is not on the Goldbach problem. It is now generally accepted in the mathematical community that this concept first appeared when Hardy and Ramanujan studied the problem of "asymptotic analysis of integer splitting", and then when Hardy and Littlewood cooperated to study Waring's problem , is completed.

Today, as an important tool for studying Goldbach's conjecture, this tool has been carried forward by later generations of mathematicians.

For example, Helfgott standing on the podium is the master of round method theory in the field of number theory today.

"...The connotation of Goldbach's conjecture is that any even number greater than 2 can be written as the sum of two prime numbers. Let's call it conjecture A."

"...Because an odd number minus an odd prime number is an even number, conjecture A thinks that any even number is equal to the sum of two prime numbers, so conjecture A can be used to infer conjecture B, any odd number greater than 9 can be written as the sum of three odd prime numbers .”

At this point in the opening remarks, Helfgott paused and continued.

"And the 'round method' I described is a weak conjecture that proves Goldbach's conjecture, that is, conjecture B!"

Conjecture A is true, and conjecture B must be true.

But the other way around, it doesn't work.

As for why, this involves a very interesting question in logic mathematics. It is difficult to describe it with elementary mathematics, but if it is explained in descriptive language, it is a set composed of "the sum of any odd number greater than 9 and an odd prime number", which is not equivalent to the set "any even number", and the intersection of All elements are infinitely many, and cannot be proved exhaustively.

In fact, from an abstract point of view, whether it is the "even-numbered set" of the circle method or the "1+1 form" of the sieve method, everyone is in the same category, and they are all short of the final touch.

This distance may be separated by a river, or it may be two mountains facing each other.

After a short opening remark, Helfgott didn't talk nonsense, and wrote a line of calculations on the whiteboard.

[...When 2||N, there is r3(N)=1/2n(N/N)∏(1-1/(p-1))∏(1+1/(p-1)), (1 +O(1))]

The moment Lu Zhou saw this line of calculation, his eyes lit up slightly.

This line of expression was not written by the old man casually, it was one of the many expressions proposed in the 1922 paper by Hardy and Littlewood, two big figures in the field of digit theory!

When studying the twin prime number conjecture, Lu Zhou happened to consult that document, and even quoted some of its conclusions.

It is precisely because of this that he can be said to be very impressed with this.

It seems that this report will be interesting.

The old man standing in front of the whiteboard continued to write with a marker pen without saying a word.

There was no sound in the venue.

Not only Lu Zhou listened very seriously, but even the other big bosses listened very carefully.

There is a specialization in the art industry, even a big shot, it is impossible to penetrate into other people's field in an instant. Therefore, the papers at the general report meeting will be released in advance on the official website of the meeting for people to preview, and write down the questions to be asked in the notes.

If the report does not answer your own questions, ask questions during the questioning session. This is the correct posture for listening to an academic report.

More than forty minutes passed, Helfgot stopped the marker pen in his hand, turned to look at the venue.

"The basic proof process is like this. If you have any questions, you can ask them now."

Lu Zhou raised his hand.

Helfgott and Lu Zhou met their gazes, nodded, and motioned for him to get up and speak.

After glancing at the notes, Lu Zhou stood up and asked.

"I have doubts about the formula listed in your line 34. In your operation of =∑a(n)z^n+δ(n), you can directly get every integer n>0. I guess you may use It is the Cauchy-Gusa theorem or its corollary residue theorem. But how do you judge that the function f(s) is a holomorphic function?"

There were murmurs of discussion in the hall.

Obviously, the question Lu Zhou asked touched the hearts of many people.

"That's a good question," Helfgott glanced at Lu Zhou unexpectedly, turned around and wrote a line of calculations on the whiteboard, and then tapped on it with a marker, "Do you understand?"

Seeing the calculation formula, Lu Zhou's expression was slightly stunned, and he nodded.

"Understood, thank you."

Nodding politely, Lu Zhou sat back and copied the supplementary formula on the whiteboard into his notebook.

Although his research is mainly on the sieve method, Mr. Helfgott's method is also very inspiring to his research work. The so-called research work is exactly like this, perfecting one's own theories in exchanges and discussions, and rubbing out new viewpoints in the collision of thinking.

While Lu Zhou was sorting out his notes, someone next to him lightly poked his arm.

"Excuse me, can I ask you a question?"

The one who spoke was a little girl with slightly pale complexion and slightly curly blond hair.

The reason why I say it is a little girl is because she looks young, and she is a little shorter than Lu Zhou. She is probably an undergraduate student at UC Berkeley... If you say she is a graduate student, Lu Zhou will never believe it.

Although her English pronunciation is a bit jerky, her voice is very soft, which is surprisingly nice.

Regardless of whether the voice is good or not, for Lu Zhou, if someone discusses a math problem with him, as long as it is not unreasonable, he will never refuse, so he said generously: "Ask."

The girl blinked, pointed at the whiteboard with some embarrassment, and said, "I'm sorry, that... what did you understand just now?"

Seeing that line of calculation, she didn't understand it at all.

"Are you asking about that expression?" Probably guessing the question she wanted to ask, Lu Zhou patiently explained: "Because I(n)=∫{f(s)/s^(n+ 1)}ds=2πian, this is a closed-orbit integral, so returning to the original formula, you can directly use the residue theorem. Professor Helgoft's explanation may be jumpy, and it is really difficult to understand. Think more .”

Listening to Lu Zhou's explanation, the girl panickedly took notes in her notebook.

From her meticulous note-taking technique, Lu Zhou was even more convinced of his guess that she was probably studying for an undergraduate degree.

But can an undergraduate really understand this kind of lecture?

Afraid that she would be embarrassed to ask, Lu Zhou said casually, "Any questions?"

"Thank you, no more... Excuse me, can you give me your email address? I still have a lot of questions to ask... you." Because she was too nervous, this girl who seemed a bit rash accidentally bit her tongue, His face turned red.

It can be seen that she is not very good at communicating with people.

Also not particularly good at communication, Lu Zhou can understand it, so he didn't care, and said casually: "It's okay. Besides, you don't have to always say 'I'm sorry'. My name is Lu Zhou, what is your name?"

"I know your name is Lu Zhou, and I met you at the opening ceremony," the girl added embarrassingly, perhaps because she suddenly remembered that she hadn't reported her name, "My name is Vera, and I study in Berkeley... ...interested in pure mathematics, especially number theory.”

Vera?

Sounds a bit like Hebrew, Russian?

Lu Zhou subconsciously glanced at her chest. Although it was not flattering, it was indeed a bit shabby.

emmm...

Probably not?

"May I ask, how old are you this year?"

"17..."

Lu Zhou looked at her in surprise: "Can I go to Berkeley at the age of 17?"

At this age, he has not yet graduated from high school.

"I'm recommended by the IMO Gold Medal..." Vera smiled shyly, and said with admiration, "Of course, it's nothing compared to you who have already solved two mathematical conjectures..."

Lu Zhou was stunned for a moment and said: "...No, the gold medal in the Olympic mathematics competition is already very strong. You can be more confident and don't belittle yourself. Surprisingly, you won the gold medal at the age of 15? Then how old were you? high school—”

At this moment, the last questioner finished his speech, and seeing that no one continued to ask questions, Mr. Helfgott on the stage announced the end of the report meeting.

"We are still a long way from fully proving Goldbach's conjecture."

"This is the end of my report, thank you all for coming!"

Helfgott nodded slightly, and walked down the stage amid applause.

Because he has not participated in the IMO competition, Lu Zhou is still very interested. Originally, he planned to chat with the little girl who won the gold medal, but seeing that it was getting late, and he happened to have something to do, so he put it aside, packed up his notes, and walked out of the venue go.