Volume 1: Student Life Chapter 274: Didn’t you catch him?

The document that caught Chen Zhou's eye was another .

That is, the circular method.

It and the sieve method have always been the two most important methods in the field of number theory research.

Of course, in addition to the sieving method and the circle method , there are also methods such as density ratio.

The full name of the method is Hardy-Littlewood-Ramanujan method.

The names mentioned are British mathematician Hardy, British mathematician Littlewood and Indian mathematician Ramanujan.

Chen Zhou was familiar with any of these three people.

Ramanujan's outstanding contribution to mathematics is so great that in India, he is called the "Son of India" along with Mahatma Gandhi, poet Tagore and others.

Moreover, there are now two international mathematics awards named after Ramanujan.

Hardy and Littlewood, both British mathematicians, made outstanding research in Diophantine analysis, heap number theory, multiplicative number theory, trigonometric series, etc.

And together they completed a new proof of Waring's theorem.

Speaking of trigonometric series, Fourier series is a type of trigonometric series.

As for the relationship between the three, in Hardy's words, his greatest achievement in mathematics was "the discovery of Ramanujan."

It was with the help of Hardy that Ramanujan gradually emerged as a mathematician.

Speaking of Hardy.

In a sense, it can be said that he influenced the thinking of a generation of Chinese mathematicians.

The reason why China has achieved the level of "1+2" in number theory, or in the Goldbach conjecture, was achieved by Mr. Chen.

In fact, it has some connection with Hardy.

Mr. Chen’s teacher was Mr. Hua, and Mr. Hua’s teacher was Hardy.

It’s just that the method Mr. Chen used to advance the Goldbach conjecture to “1+2” was the weighted sieve method, not the circle method.

The circle method was originally invented by Hardy and Littlewood when they were working on the theory of prime numbers.

Then, they found that this thing seemed to have some connection with the Goldbach conjecture.

So we perfected the theory of the circle method and gave a method, a method to describe the "decomposition method" in mathematical language.

That is, through the iconic integral formula of the circle method.

【∫01e^（2πimα）dα】

Consider this integral, when m=0, ∫01e^0dα=1.

When m≠0, the exponent cannot be 0. According to Euler's formula, the entire power becomes 0.

So the whole integral is 0.

Using this property, we can transform the integral into a decomposition function.

Every decomposition method of N=p1＋p2, p1, p2≥3 can be written as D(N)=∫01(2＜p≤N∑e^(2πiαp)^2)e^(2πi α(-N))dα.

Similarly, the decomposition of N=p1＋p2＋p3, p1, p2, p3≥3 can be written as T(N)=∫01(2＜p≤N∑e^(2πiαp)^3)e^(2πiα(-N))dα.

In this way, to prove [there is always a splitting method] is to prove that for any N that satisfies the question, D(N)>0 and T(N)>0.

At this point, we can start discussing points.

This is the main idea of the Circle Method.

The essence of the circle method is Fourier analysis applied in number theory.

Simply put, it is to analyze the functions on the circle.

In contrast, the sieve method, like the heads and tails of a coin, aims to give an approximate estimate of the distribution of prime numbers.

"Since the sieving method may not work, let's try the circle method..."

Chen Zhou was thinking about it in his mind, but his hands were not in a hurry.

He began to search for literature related to the Circle Method.

If you want to do your work well, you must first sharpen your tools.

Chen Zhou has not yet fully understood the application of the circle method.

Not to mention, it will be used immediately to solve the correction problem of Cramer's conjecture.

Chen Zhou's eyes were unusually bright, with a hint of expectation in them.

He stared closely at the computer screen in front of him, absorbing the knowledge content on it to enrich his own knowledge.

In fact, in addition to the sieve method and the circle method, there are many other tricks in the field of number theory.

For example, the generalized Riemann hypothesis can be used to prove some limited special cases.

Then use these special cases to prove something else.

It's like the so-called "zero-free zone".

Although I still don’t know how to prove that the real part of all non-trivial zeros is 1/2.

However, it has been proved that the zero point must be in a region containing the so-called "critical line", and this region is very small near the real axis.

Since then, people have been using similar conclusions to prove other problems.

However, Chen Zhou doesn't like this method very much.

Because he always felt it was a bit strange to use an unproven conjecture to solve another conjecture.

What if the Riemann hypothesis is falsified?

Even though the probability is very small, and even though thousands of mathematical problems have been solved based on the Riemann hypothesis, Chen Zhou is still unwilling to try.

He still hopes to take every step firmly.

Of course, if one day he can prove the Riemann hypothesis.

That's another matter.

Time passed slowly, and after reading several papers, Chen Zhou started to practice.

Yang Yiyi on the side looked at what Chen Zhou wrote on the draft paper with some curiosity.

However, after reading it once, she didn't quite understand it.

Yang Yiyi naturally had no intention of studying it in depth, she was just attracted by Chen Zhou's state.

This situation is a bit familiar...

How should I put it? It's like...

It was just like the feeling last time when Chen Zhou was about to solve the hail conjecture.

Could it be?

Yang Yiyi thought so with a hint of surprise in her eyes.

She remembered that she had heard Chen Zhou say last time that he was studying the Cramer conjecture, which seemed to be a difficult mathematical problem that had troubled the mathematical community for nearly a hundred years, right?

Is it going to be resolved so soon?

Yang Yiyi just looked at Chen Zhou, a little dazed for a moment.

Chen Zhou is fully studying how to use the circle method to solve the correction problem of Cramer's conjecture.

While reading the literature, there was a moment when he felt that he had caught a fleeting inspiration.

However, as time went on, he felt that this problem was really difficult...

There is no doubt that he did not seize the inspiration of that moment.

He also did not succeed in solving this correction problem.

Slowly, the speed at which the pen in Chen Zhou's hand rubbed against the draft paper slowed down.

Chen Zhou's originally bright eyes also became a little confused.

His eyebrows were already wrinkled unconsciously.

"Alas..." Chen Zhou sighed lightly and finally stopped writing.

I just held the pen out of habit and kept tapping on the draft paper.