Chapter 213: 1 small step to the top

The full name of the circle method is the Hardy Littlewood 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 was not on Goldbach. It is now generally accepted in the mathematical community that this concept was first introduced by Hardy in the study of "asymptotic analysis of integer splitting" with Ramanugin, and then when Hardy collaborated with Litewood on the Waring problem , Was added intact.
Today, as an important tool for studying Goldbach's conjecture, this tool has been carried forward by later mathematicians.
For example, Helfgot, who is standing on the podium, is the big bull in the circle 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 odd numbers minus odd prime numbers are even numbers, conjecture A thinks that any even number is equal to the sum of two prime numbers, so conjecture A can be deduced to guess B, and any odd number greater than 9 can be written as the sum of three odd prime numbers. . "
With the opening remarks here, Helf Gott paused and continued.
"And the" round method "I'm talking about is a weak conjecture that proves his Goldbach 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 an interesting question in logical mathematics. It is difficult to describe in elementary mathematics, but if it is explained in descriptive language, it is a set consisting of "the sum of any odd number and odd prime number greater than 9", which is not equivalent to the set of "any even number" and intersects There are infinitely many elements, and there is no exhaustive proof.
In fact, from an abstract point of view, whether it is the "even set" of the circle method or the "1 + 1 form" of the sieve method, everyone is half-pounded, and they are all in the end.
This distance may be separated by a river or two mountains.
After a brief opening, Helfgot didn't talk nonsense, and wrote a line of calculations on the whiteboard.
[... when 2
N, r3 (N) = 1 / 2n (N2 / N3) ∏ (1-1 / (p-1) 2) ∏ (1 + 1 / (p-1) 2), (1 + O (1))]
The moment he saw this formula, Lu Zhou's eyes lightened slightly.
This line of expression is not written by the old man, it is one of many expressions proposed by Hardy and Littlewood in the 1922 paper!
When researching the twin prime conjecture, Lu Zhou happened to consult that document and even cited some of its conclusions.
It is for this reason that he can be said to be very impressed.
It seems this report will be interesting.
The old man standing in front of the white board said nothing, and continued to write wryly with a marker.
There was no sound at the venue.
Not only did Lu Zhou listen very carefully, but even the other big brothers listened very carefully.
There is a specialization in the surgery industry, and even a big man can't go deep into other people's fields in an instant. Therefore, the papers in the general report will be released in advance on the conference website for people to preview, and the questions to be asked will be written on the notes.
If the report does not answer your own questions, ask questions during the questioning session. This is the correct posture to listen to the academic report. It is not just to watch a lively and applause in the past.
Forty minutes later, Helfgot stopped the marker in his hand and turned to look at the venue.
"This is the basic proof process. If you have any questions, you can ask questions now."
Lu Zhou raised his hand.
Helfgot and Lu Zhou met their eyes, nodded, and signaled that he could get up and speak.
Glancing at the notes, Lu Zhou stood up and asked.
"I have doubts about the formulas listed on your line 34. In your calculation of = ∑a (n) z ^ n + δ (n), you directly get each integer n> 0. I guess you might It's the Cauchy-Gusa theorem or its inferred remainder theorem. But how do you judge that the function f (s) is a holomorphic function? "
A whisper was heard in the venue.
Obviously, Lu Zhou asked this question to the hearts of many people.
"That's a good question," Helfgot said, unexpectedly glancing at Lu Zhou, turning to write a line of calculations on the whiteboard, and then tapping on the marker, "Got it?"
Seeing that line of expressions, Lu Zhou looked slightly stunned and nodded.
"I see, thank you."
Nodded politely, Lu Zhou sat back, and copied the line of supplementary calculations on the whiteboard to his notebook.
Although his research is mainly on sieving, Mr. Helfgot's method is also inspiring to his research work. The so-called research work is exactly the same, perfecting their theories in communication and discussion, and rubbing out new ideas in the collision of thoughts.
Just as Lu Zhou was finishing his notes, someone beside him gently poked at his arm.
"Sorry, can I ask you a question?"
It was a little pale girl with a little curly blond hair.
The reason why it is a little girl, because she looks young, is a little shorter than Lu Zhou, probably an undergraduate at Berkeley ... If she is a graduate student, Lu Zhou is absolutely unbelievable.
Although her English pronunciation is a bit jerky, her voice is very light and unexpectedly a bit nice.
Regardless of whether the sound is good or not, for Lu Zhou, someone discusses mathematical problems with him. He will never refuse as long as it is not unreasonable, so he said generously: "Ask."
The girl blinked and pointed a little awkwardly on the whiteboard and said, "Sorry, that ... what did you just understand?"
Seeing that line of formula, she didn't understand it at all.
"You asked that expression?" I guessed the question she wanted to ask, and Lu Zhou explained patiently: "Because I (n) = ∫ {f (s) / s ^ (n + 1)} ds = 2πian, this is a closed-orbit integral, so back to the original formula, you can directly use the remainder theorem. Professor Helgoft's idea may be jumping, it ’s really difficult to understand, think about it . "
Listening to the explanation of the landing boat, the girl hurriedly took notes in the book.
From her minute note-taking technique, Lu Zhou was more convinced of her guess, and she was probably studying for an undergraduate degree.
But can undergraduates really understand such lectures?
Afraid she was embarrassed to ask, Lu Zhou casually said, "Do you have any questions?"
"Thank you, no ... Sorry, can you give me your e-mail? I have a lot of questions to ask ... you." Because she was too nervous, the girl who looked a bit offended accidentally bit her tongue. His face flushed.
It can be seen that she is not very good at communicating with people.
It is not particularly good at communication, but Lu Zhou can understand, so he didn't care, and said casually: "It doesn't matter. In addition, you don't always have to say 'I'm sorry.'
"I know your name is Lu Zhou ~ EbookFREE.me ~ I met you at the opening ceremony," it may have suddenly occurred to me that I hadn't reported my name yet, and the girl added embarrassedly. La, studying at Berkeley ... interested in pure mathematics, especially in number theory. "
Willa?
Sounds a bit like Hebrew, Russian?
Lu Zhou glanced subconsciously at her chest. Although it wasn't a flat place, it was indeed shabby.
emmm ...
Probably not?
"Ask me how old are you this year?"
"17 ..."
Lu Zhou gave her a surprised look: "Can I go to Berkeley at 17?"
At this age, he hasn't graduated from high school yet.
"I'm an IMO gold medal ..." Vera smiled embarrassedly, said with a little admiration, "Of course, compared with you who have already solved two mathematical conjectures ..."
Lu Zhou stunned and said, "... No, the gold medal in the Olympic Mathematical Competition is already very strong. You can be confident and not arrogant. Surprisingly, you won the gold medal at 15? How old are you? High school— "
At that moment, the last questioner finished his statement and saw that no one continued to ask questions, and Mr. Helfgot on the stage announced the end of the report.
"We still have a long way to go to fully prove Goldbach's conjecture."
"That's it for my presentation, thank you for coming!"
Helfgort nodded slightly and walked off the stage in applause.
Because he had not participated in the IMO competition, Lu Zhou was still very interested. He originally planned to talk to the little girl who won the gold medal, but seeing that it was too late, and he happened to have something to do, so he set it aside, picked up his notes, and walked outside the venue. go with.