[1] THE SCHOLAR ADVANCED TECHNOLOGY SYSTEM (END)

[1] THE SCHOLAR ADVANCED TECHNOLOGY SYSTEM (END)
Chapter 213: One Small Step

Translator: Translated Henyee Editor:Translated Henyee


The full name of the circle method is "Hardy-Littlewood Circle Method". It was not only an important tool for Goldbach's conjecture but also an important tool for analytic number theory.


The purpose of using this tool is not necessarily for Goldbach's alleged use. It is now widely believed in the mathematical analysis community that this concept first appeared in Hardy's research on "the analysis of integer cleavage symptoms". When Hardy and Littlewood collaborated on Hualin's problem, this method was fully completed.


As an important tool for studying Goldbach's conjecture, this method has been developed by other mathematicians.


For example, Helfgott who stood on stage was one of the contributors to the circle method.


"... Goldbach's conjecture is that even numbers greater than 2 can be written as the sum of two prime numbers. We can call it a guess. ”


“... Since an odd number minus an odd prime number is an even number, guess A thinks that an even number equals the sum of two primes. Therefore, guess B can be used to guess inference B. Any odd number greater than 9 can be written as the sum of three odd primes. ”


Helfgott paused for a moment before continuing, "'The circle method' I am talking about is a weak conjecture that proves part of Goldbach's conjecture, guess B!"


Only if guess A is established, guess B will also occur.


However, this will not work otherwise.


As for why, it is because it involves a very interesting question on logical mathematics. It is difficult to explain with simple mathematics, but basically a set of "numbers of odd and odd primes greater than 9" is not equivalent to a set of "even numbers".All elements are infinite and cannot be proven in depth.


From an abstract point of view, the "even set" of the circle method is the "1 +1" form of the sieve method. There was a small part missing in both.


However, this small part is very important.


After a brief opening comment, Helfgott began writing a series of calculations on the board.


[... when 2 || N, there is r3 (N) \= 1 / 2n (N2 / N3) ∏ (1-1 / (p-1) 2) ∏ (1 + 1 / (p-1) 2), (1 + O (1))]


Lu Zhou's eyes twinkled when he saw this line of calculations.


These expression lines are not just doodling. It was a two-digit argument from Hardy and Littlewood. That is one of the expressions presented in the 1922 thesis!


While studying the main guess of twins, Lu Zhou read the thesis. He even cited some passages in his own thesis.


Therefore, the effect on this thesis is very profound.


This report seems a little interesting.


The old man in front of the blackboard did not speak. Instead, he continued to write.


The place was completely quiet.


It was not only Lu Zhou who listened carefully. All the other big names were also listening seriously.


The mathematics industry is highly specialized. No one is an expert in everything. Therefore, the thesis for this report will be released in advance for everyone to study and consult.


If the report does not answer someone's question, people will be able to ask questions during the FAQ section. This is how academic reporting is done. One should actively think and ask questions and participate in discussions.


After 40 minutes, Helfgott finally stopped writing and turned around.


“Basic proving process like this.If you have any questions, you can ask now. ”


Lu Zhou raised his hand.


Helfgott looked at Lu Zhou and nodded.


Lu Zhou stood up and asked, "I doubt the formula on line 34. In operation \= (n) z ^ n + ⁇ (n), you can directly get each integer n> 0. I guess you use the Cauchy-Gusa theorem or its inferential residue theorem.But how do you judge that the function f is a pure function? "


A quiet discussion began at the venue.


Clearly, Lu Zhou's question was interesting.


"Good question," Helfgott said as he looked at Lu Zhou. He then wrote down a row of calculations on the blackboard before he asked, "Do you understand now?"


Lu Zhou stared at the calculation line and nodded.


"Understand me, thank you."


Since his main research is on sieve theory, the Helfgott method is also interesting. By making an academic exchange, Lu Zhou was able to refine his own theories and use dissent as a way to gain inspiration.


While Lu Zhou was making notes, the person next to him patted his arm.


"I'm sorry, can I ask you a question?"


The one who asked the question was a blonde-haired girl with pale complexion.


This girl looked young and she was slightly shorter than Lu Zhou. He's probably an undergraduate student from Berkeley.


Her voice is pleasant to hear.


Despite the pleasure of the voice, Lu Zhou would never reject a mathematical question. He said, "Please."


The girl blinked and pointed to the blackboard as she asked, "Sorry, that's ... What do you know about that?"


He looked at the formula line that he did not understand at all.


"You're talking about expression?" Ask Lu Zhou. He then patiently explained, “Because I (n) \= ∫ {f (s) / s^ (n + 1)} ds \= 2 is a closed loop integral, the, You can use the residue theorem directly when you return to the original form .Professor Helfgott's explanation is somewhat funky, so it's hard to understand. Just think about it more. ”


The girl started writing notes.


From his cruel note-taking technique, Lu Zhou was convinced that this girl was a student.


However, can a scholar really understand this report?


Lu Zhou asked, "There's another question?"


"Thank you, no ... Sorry, can you give me your email? I've got a lot of questions to ask you, ” said the girl.She looked a little nervous and started blushing.


It was obvious that he was not good at socializing.


Lu Zhou was also not good at socializing, so he did not care and said, "Surely, don't say "sorry" all the time. I'm Lu Zhou, and you? "


"I know you are Lu Zhou. I saw you at the opening ceremony, ” said the girl. He then said, “I Vera. I studied at Berkeley ... I am very interested in pure mathematics, especially number theory. ”


Vera's?


Sounds a bit Russian?


Lu Zhou unconsciously looked at ***********. Although they are not the size of a washboard, they are on the smaller end.


Emm ...


Not likely?


"Just out of curiosity, how old are you?"


"17 ..."


Lu Zhou looked at him and asked, "A 17-year-old can attend Berkeley?"


He didn't even graduate High School when he was 17.


"I'm an IMO 1 gold medalist." said Vera. He smiled and said, "Of course, that's nothing compared to finishing two conjectures ..."


Lu Zhou said, “... No, the Olympic Math Competition is impressive. More confident. It's unsurprising. So, you got a medal when you were 15? When did you go to that high school? "


The last question was not answered by Vera when Helfgott announced the end of the report.


"We still have a long way to go to prove Goldbach's conjecture."


"Thank you for coming!"


Helfgott then nodded and walked down the stage in a standing ovation.


Lu Zhou had never participated in an IMO competition before, so he was quite interested. He wanted to talk to this girl for a while, but it was too late. Therefore, he packed up his things and started walking out of the venue.