Chapter 213: One Small Step
Translator: Henyee Translations Editor: Henyee Translations
The full name of the circle method was “Hardy–Littlewood circle method”. It was not only an important tool for the Goldbach’s conjecture but also an important tool for a.n.a.lytical number theory.
The intended use of this tool was not necessarily for Goldbach’s conjecture. It was now widely believed in the mathematical a.n.a.lysis community that this concept first appeared in Hardy’s research on “symptomatic a.n.a.lysis of integer splitting”. When Hardy and Littlewood collaborated on the Hualin issue, this method was fully completed.
As an important tool for studying Goldbach’s conjecture, this method had been expanded by other mathematicians.
For example, Helfgott who stood on stage was one of the contributors to the circle method.
“… The meaning of the Goldbach’s conjecture is that any even number greater than 2 can be written as the sum of two prime numbers. We can call this guess A.”
“… Because the odd number minus the odd prime number is an even number, guess A thinks that any even number is equal to the sum of the two prime numbers. Therefore, guess B can be used to guess the inference 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, “The ‘circle method’ I’m talking about is the weak conjecture that proves part of the Goldbach’s conjecture, the guess B!”
Only if guess A was established, would guess B be established as well.
However, this would not work the other way around.
As for why, it was because this involved a very interesting question on logical mathematics. It was difficult to describe with simple mathematics, but it was basically a set of “the sum of odd and odd primes greater than 9” was not equivalent to the set of “any even numbers”. All elements were infinite and could not be proved exhaustively.
From an abstract point of view, the “even set” of the circle method was the “1+1” form of the sieve method. There was a small part missing in both.
However, this small part was crucial.
After a brief opening remark, Helfgott started to write a line of calculations on the whiteboard.
[… 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 lit up when he saw this line of calculations.
This line of expression was not merely scribbling. It was the two-digit argument of Hardy and Littlewood. It was one of the expressions that were presented in the 1922 thesis!
While studying the twin prime conjecture, Lu Zhou read that thesis. He even quoted some parts in his own thesis.
As such, his impression of this thesis was deep.
It seems that this report is a bit interesting.
The old man in front of the whiteboard did not speak. Instead, he continued to write.
The venue was completely quiet.
It was not just Lu Zhou who was listening carefully. All of the other big names were also listening seriously.
The mathematics industry was highly specialized. No one was an expert at everything. Therefore, the thesis for the report would be released in advance for everyone to study and consult.
If the report did not answer one’s question, one would be able to ask the question during the Q&A section. This was how academic reports were done. It was not just watching and listening. One had to actively think and ask questions as well as to partic.i.p.ate in discussions.
After 40 minutes, Helfgott finally stopped writing and turned around.
“The basic proof process is like this. If you have any questions, you can ask them now.”
Lu Zhou raised his hand.
Helfgott looked at Lu Zhou and nodded.
Lu Zhou stood up and asked, “I have doubts on the formula on line 34. In the operation of =∑a(n)z^n+δ(n), you can directly derive each integer n>0. I guess you used the Cauchy-Gusa theorem or its inference residue theorem. But how do you judge that the function f(s) is a pure function?”
Quiet discussions began in the venue.
Clearly, Lu Zhou’s question was intriguing.
“Good question,” said Helfgott as he looked at Lu Zhou. He then wrote down a line of calculations on the whiteboard before he asked, “Do you understand now?”
Lu Zhou looked at the line of calculations and nodded.
“Understood, thank you.”
Lu Zhou sat back down and copied the line of formula into his notebook.
Since his main research was on sieve theory, Helfgott’s method was also interesting. By doing academic exchanges, Lu Zhou could perfect his own theory and used the difference in opinions as a way of getting inspiration.
While Lu Zhou was taking notes, someone next to him poked his arm.
“Sorry, can I ask you a question?”
The person that asked the question was a blonde girl with pale skin.
This girl looked young and she was a little shorter than Lu Zhou. She was probably an undergraduate student from Berkeley.
Her voice was pleasant to listen to.
Regardless of the pleasantness of the voice, Lu Zhou would never reject a mathematics question. He said, “Go ahead.”
The girl blinked and pointed at the whiteboard as she asked, “Sorry, that… What did you know from that?”
She looked at the line of formula which she did not understand at all.
“You’re talking about the expression?” asked Lu Zhou. He then patiently explained, “Because I(n) = ∫{f(s)/s^(n+ 1)}ds=2πian is a closed-loop integral, you can use the residue theorem directly when you return to the original form. Professor Helfgott’s explanation is a bit funky, so it is hard to understand. Just think about it more.”
The girl started to write notes.
From her ruthless note-taking technique, Lu Zhou was convinced that this girl was an undergrad.
However, could an undergrad really understand this report?
Lu Zhou asked, “Any other questions?”
“Thanks, no… Sorry, can you give me your email? I have more questions to ask you,” said the girl. She looked a little nervous and she ever started to blush.
It was obvious that she was not that good at socializing.
Lu Zhou was not that good at socializing either, so he did not care and said, “Sure. Also, don’t say “sorry” all the time. I’m Lu Zhou, and you are?”
“I know you’re Lu Zhou. I saw you at the opening ceremony,” said the girl. She then said, “I’m Vera. I’m studying at Berkeley… I’m very interested in pure mathematics, especially number theory.”
Vera?
Sounds a bit Russian?
Lu Zhou subconsciously looked at her b.o.o.bs. Although they were not washboard size, they were on the smaller end.
Emm…
No way?
“Just out of curiosity, how old are you?”
“17…”
Lu Zhou looked at her and asked, “A 17-year-old can attend Berkeley?”
He had not even graduated from high school when he was 17.
“I’m an IMO 1 gold medalist…” said Vera. She smiled and said, “Of course, it’s nothing compared to solving two conjectures…”
Lu Zhou said, “… No, the Olympic Math Compet.i.tion is impressive. Have more confidence in yourself. This is shocking. So you got the medal when you were 15? When did you go to high school then?”
The last question was left unanswered by Vera as Helfgott announced the end of the report.
“We still have a long way to go to prove Goldbach’s conjecture.”
“Thanks for coming!”
Helfgott then nodded and walked down the stage in the round of applause.
Lu Zhou had never partic.i.p.ated in the IMO compet.i.tion before, so he was quite interested. He wanted to talk with this girl for a bit, but it was getting late. Therefore, he packed up his stuff and started to walk out of the venue.