The full name of the circle method was "Hardy Littlewood circle method". Not only was it an important tool for studying Goldbach's conjecture, but it was also an important tool often used in analytic number theory.
The invention of this tool was not based on Goldbach's conjecture. The general view of the mathematics community was that this concept first appeared when Hardy and Ramanujan were researching the "asymptotic analysis of integer splitting" problem. Then, when Hardy and Littlewood collaborated on Waring's conjecture problem, it was completed.
Today, as an important tool for studying Goldbach's conjecture, this tool had been carried forward by later generations of mathematicians.
For example, Helfgott, who was standing on the podium, was a big name in today's number theory field.
"… 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 is an even number, conjecture A states that any even number is equal to the sum of two prime numbers. Therefore, conjecture A can be used to deduce conjecture B. Any odd number greater than 9 can be written as the sum of three odd prime numbers."
Helfgott paused for a second before he continued.
"And the 'circle method' I talked about is to prove the weak conjecture of Goldbach's conjecture, which is conjecture B!"
If conjecture A was true, then conjecture B must be true.
But it was not the other way around.
As for why, it involved a very interesting problem in logic and mathematics. It was difficult to describe with elementary mathematics, but to explain it in descriptive language, it was the set formed by "the sum of any odd and odd prime numbers greater than 9", which was not equivalent to the set formed by "any even numbers", and all the elements in the intersection set were infinite, so it could not be proved exhaustively.
In fact, from an abstract point of view, be it the circle method's "even set" or the sieve method's "1 + 1 form", they were both about the same. They both lacked the final step.
This distance could be separated by a river, or it could be two mountains facing each other.
After a brief introduction, Helfgott didn't waste any time and wrote down a line of calculations on the whiteboard.
[When 2 | | N saw this line of calculations, Lu Zhou's eyes lit up.
This line of expression was not written randomly by the old man. It was one of the many expressions proposed by Hardy and Littlewood in the 1922 paper!
When Lu Zhou was researching the twin prime number conjecture, he happened to read that paper. He even quoted some of the conclusions in it.
It was also because of this that he had a deep impression of this.
Looks like this report is going to be interesting.
The old man standing in front of the whiteboard did not say a word and continued to write with a marker.
The venue was dead silent.
Not only was Lu Zhou listening attentively, but the other big shots were also listening attentively.
Everyone had their own specialties. Even a big shot would not be able to enter into someone else's domain in an instant. Therefore, the thesis for the report would be posted on the conference's official website in advance for people to preview and write down the questions they were going to ask in their notes.
If the report did not answer one's question, one should ask the question during the Q&A session. This was the correct way to listen to an academic report. One should not just go and watch the show and clap their hands.
After more than forty minutes, Helfgott stopped writing and turned around to look at the venue.
"This is the basic proof process. If you have any questions, you can ask them now."
Lu Zhou raised his hand.
Helfgott and Lu Zhou looked at each other and nodded, indicating that he could stand up and speak.
Lu Zhou looked at the notes and stood up. He then spoke.
"I have questions about the equations you listed on line 34. You can directly get every integer n > 0 from the operation = ∑ a (n) z ^ n + δ (n). I guess you might be using the Cauchy-Goussa theorem or its corollary residue theorem. But how did you determine that the function f (s) is a holomorphic function? "
There was a lot of discussion in the venue.
Obviously, Lu Zhou's question struck a chord in many people's hearts.
"That's a good question," said Helfgott as he looked at Lu Zhou. He then turned around and wrote down a line of equations on the whiteboard. He then tapped on the whiteboard and asked, "Do you understand?"
When Lu Zhou saw the line of equations, he nodded.
"I understand, thank you."
Lu Zhou nodded politely and sat back down. He then copied the line of equations on the whiteboard into his notebook.
Although his research was mainly on sieve methods, Helfgott's method was also a great inspiration for his research work. The so-called research work was like this. Perfect one's own theory through discussion and create new ideas through the collision of ideas.
While Lu Zhou was organizing his notes, someone next to him gently poked his arm.
"Sorry, can I ask you a question?"
The person who spoke was a girl with pale skin and slightly curly blonde hair.
The reason why he said it was a girl was because she looked young. She was a bit shorter than Lu Zhou. She was probably an undergraduate student at UC Berkeley … Lu Zhou would never believe that she was a graduate student.
Although her English pronunciation was a bit shaky, her voice was soft and surprisingly pleasant to the ears.
Regardless of whether her voice was pleasant to the ears or not, Lu Zhou would never refuse to discuss a mathematics problem with someone. Therefore, he said, "Go ahead."
The girl blinked and awkwardly pointed at the whiteboard and said, "Sorry, um … What did you just understand?"
She looked at the line of equations and did not understand at all.
"You're asking about that expression?" Lu Zhou probably guessed what she wanted to ask, so he patiently explained, "Because I (n) in the linear equation, I (n) = ∈ {f (s)/s ^ (n + 1)} ds = 2πian. This is a closed-orbit integral. Therefore, you can use the residue theorem. Professor Helfgott's explanation may be a bit jumpy, so it's really hard to understand. You have to think about it more. "
The girl listened to Lu Zhou's explanation as she nervously took notes in her notebook.
From the way she took notes, Lu Zhou was even more certain of his guess. She was probably studying for an undergraduate degree.
However, can an undergraduate really understand a lecture like this?
Afraid that she would be embarrassed to ask, Lu Zhou casually said, "Do you have any other questions?"
"Thank you, no … Sorry, can you give me your email? I still have a lot of questions to ask … you. " Because she was too nervous, this seemingly rash girl accidentally bit her tongue, causing her face to turn red.
It could be seen that she was not good at communicating with people.
Lu Zhou was not good at communicating either, so he did not mind and said, "It's okay. Also, you don't have to say 'sorry' all the time. My name is Lu Zhou, what's your name? "
"I know your name is Lu Zhou, I saw you at the opening ceremony," said the girl. She suddenly remembered that she had not told him her name yet, so she awkwardly added, "My name is Vera, I'm studying at Berkeley … I'm very interested in pure mathematics, especially number theory."
Vera?
Sounds kind of like Hebrew. Russian?
Lu Zhou subconsciously glanced at her chest. Although it was not flat, it was a bit shabby.
Emm …
Probably not, right?
"May I ask, how old are you this year?"
"17 …"
Lu Zhou looked at her with some surprise and said, "Can a 17-year-old go to Berkeley?"
I haven't even graduated from high school yet.
"I'm an IMO gold medalist …" said Vera with an embarrassed smile. She then said in an admiring tone, "Of course, it's nothing compared to you, who has already solved two mathematics conjectures …"
Lu Zhou paused for a second and said, "No, a gold medal in the Olympic Mathematics Competition is already very impressive. You should be more confident, don't belittle yourself. Amazing, you won a gold medal at 15? Then how old were you in high school … "
The last questioner finished speaking. Seeing that there were no more questions, Helfgott announced the end of the report.
"We still have a long way to go to completely prove Goldbach's conjecture.
"That's all for my report. Thank you all for coming!"
Helfgott nodded slightly and walked off the stage amidst the applause.
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