Orion Crest, Series_1

Chapter 92: A small step to the mountaintop

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

　　The invention of this tool was not in the context of the Goldbach problem. It is now generally accepted in the mathematical community that this concept first appeared in the study of "asymptotic analysis of integer splitting" by Hardy and Ramanujan, and was later supplemented in the study of Waring's problem by Hardy in collaboration with Littlewood.

　　Nowadays, as an important tool in the study of Goldbach's Conjecture, this tool has been developed by later generations of mathematicians.

　　Helfgott, for example, standing on the podium, is today's big name in number theory, the theory of the circle method.

　　"...... Goldbach's Conjecture has the connotation that any even number greater than 2 can be written as the sum of two primes, which we shall call Conjecture A."

　　"...... Since an odd number minus an odd prime is an even number, Conjecture A assumes that any even number is equal to the sum of two primes, so using Conjecture A leads to the corollary Conjecture B, that any odd number greater than 9 can be written as the sum of three odd primes."

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

　　"And the 'circle method' I am describing is the weak conjecture that proves its Goldbach Conjecture, Conjecture B!"

　　If Conjecture A holds, Conjecture B must hold.

　　But the reverse is not true.

　　As to why, this involves an interesting problem in logical mathematics. It is difficult to describe in elementary mathematics, but in descriptive language, it means that the set consisting of "the sum of any odd number greater than 9 and any odd prime number" is not equivalent to the set of "any even number", and all the elements in the intersection set are infinitely and cannot be proved exhaustively.

　　In fact, in the abstract, whether it is the circle method of "even set" or sieve theory of "1 + 1 form", we are all the same, are short of the last foot of the door.

　　This distance may be a river or two mountains facing each other.

　　After a brief opening statement, Helfgott wrote a line of arithmetic on the whiteboard.

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

　　Orion's eyes light up slightly the moment he sees this line of arithmetic.

　　This line of expression was one of the many expressions proposed by Hardy and Littlewood, the two greats of number theory, in that 1922 paper!

　　While working on the twin prime conjecture, Orion happened to consult that paper, and even quoted some of its conclusions.

　　It was for this reason that he could be said to be deeply impressed by this.

　　It seems that this presentation, is a bit interesting.

　　The old man standing in front of the whiteboard didn't say a word, and continued to write while holding a marker.

　　The venue was silent.

　　Not only Orion was listening attentively, even the other bigwigs were also listening and watching attentively.

　　There were specialisations in different fields, and even bigwigs couldn't delve into someone else's field in an instant. That's why papers are usually released in advance on the conference website for people to study in advance and write down the questions they are going to ask on their notes.