[Let A be the set of positive integers not greater than x, and assume that P is the set of prime numbers, and then let Ap be the set of numbers in A that are divisible by the prime number p in P...]
First of all, give the definition - this can be regarded as Xiao Yi wrote this part in order to take care of his classmates who may not understand the basic elements of the sieving method.
But in fact, his classmates really knew what the screening method was, because when they took the joint entrance examination, the questions asked by the academicians included the screening method.
Even if they didn't do it, they learned about it after they went back.
Ever since, they were all stunned.
Can this problem be solved using the sieve method?
With disbelief in their eyes, they just watched Xiao Yi continue to push forward.
[Suppose z is any real number, and P(z) is the product of prime numbers in P that are not greater than z, then we have:
S(A, P, z)=|A\p|P(z)∪Ap|...]
[Consider w(n) as the number of different prime factors of a positive integer n, introduce the above formula, and after processing, we can get...]
The students in the class either became more and more confused the more they looked at it, or the more they looked at it the more surprised they became.
It seems like this problem can really be solved like this!
After gradually understanding the role of the sieving method in this question, they also gradually realized its ingenuity.
Those with broad knowledge and quick responses, such as Ye Cheng.
At this time, his expression was quite excited. "Wonderful!"
"First use the sieving method to filter out the composite terms in w(n), determine the sequence of prime terms, and process them individually, and then classify and discuss the remaining composite terms..."
"Furthermore, this method of Brother Xiao also avoids the parity problem in the sieving method. This sieving method... is definitely not the original Sieve of Eratosthenes. I visually inspected... it should be Selberg Sieve!"
Song Ziyang and Luo Qiao next to them were stunned.
How did your kid know?
Can you guess it visually?
It's not like laughing and joking every day, but secretly you're doing it better than anyone else, right?
Damn it, I must give him a good torture when I get back.
It's okay to write the book, but it's too inappropriate to write the book secretly!
And Ye Cheng never expected that he would accidentally reveal the secret rolls he usually made while explaining.
"It will be solved soon! Just continue to use the previous equivalent formula. With the help of the sieve method, w(n) is simply a little Karami!"
But at this moment, Ye Cheng's voice became confused. "Eh? Why did Brother Xiao stop?"
Luo Qiao and Song Ziyang were stunned.
Xiao Yi also stops in the process of thinking?
Is this still their Scholar Xiao?
At this time, the Xiao Yi they were looking at had just completed the classification of w(n) with Selberg Sieve, and as Ye Cheng said, it had reached the final step.
But Xiao Yi's hand left the blackboard at this time, hung down, and then looked at the content on the blackboard without moving for a long time, seemingly lost in thought.
Everyone became confused.
Some of them can already see the next steps, how could Xiao Yi not see it?
While everyone was doubting, they saw Xiao Yi's hand finally raised again. However, instead of calculating according to the steps he just answered, he started writing in the blank space on the other side.
[A(x)A(√x)(log x)^2]
[∑_(d≤y)μ^2(d)g(d)=c1·log y+c0+O((log y)^-8)]
Lines of expressions that had nothing to do with the current question appeared on the blackboard.
What followed were 30 pairs of confused eyes.
What is this?
Did they skip a certain period of time just now?
Why did he start writing the Heavenly Book without any agreement?
"Do you understand this?" Song Ziyang poked Ye Cheng, who was explaining just now, and asked maliciously.
If this kid even understands this...
"I know a lot about fairs!" Ye Cheng rolled his eyes and said angrily.
"That's good."
Then Song Ziyang breathed a sigh of relief amid Ye Cheng's confused expression.
Ye Cheng: "I can't understand that you are so happy?"
"You don't understand." Song Ziyang smiled kindly.
Liu Bin on one side of the podium may be slightly better than the students below. He can probably tell that what Xiao Yi writes is still based on the screening method, but from the perspective of the screening method, his content is more advanced.
"Is this kid...?"
As a well-informed professor, Liu Bin had some vague guesses in his mind.
But Xiao Yi, who was writing the "Book of Heaven" on the blackboard, was not so calm in his heart.
When all the inspiration that had shined before condensed, he now had enlightenment.
Maybe Longchang's enlightenment is similar?
"The biggest disadvantage of the sieving method is that there is a parity problem."
"The parity problem affects the good estimation of the sieve method for the prime number problem."
Xiao Yi was thinking in his heart.
"And just like the idea of using the sieve method to solve this problem, we need to classify, study by classification, and then break them one by one. This is undoubtedly a good idea."
"Greene related the parity problem to the difference between type I and type II information in a sieve, and his idea proved successful."
"It's just..."
The corners of his mouth curled up slightly.
"They haven't gone to the extreme yet."
The idea of classification research has a long history in the field of mathematics.
For example, he had studied p-adic Hodge theory before, and its purpose was also to classify and study a certain p-adic Gavarro representation.
Now, Xiao Yi has found the possibility to classify Twin Prime Conjecture's.
"The sifting method can be even more extreme and perfect!"
Write the last line Legendre's Formula on the blackboard, and then put a heavy period at the end.
Xiao Yi's questions were finally answered at this moment, and a smooth feeling of pleasure arose in his mind.
This is the thrill of mathematics!
Previously, the "4" that was thought to be the final difference from Twin Prime Conjecture became an insurmountable gap.
But now, he has found the theoretical material to build this bridge to the truth!
A bright smile appeared on his face.
The students in the classroom suddenly looked more and more confused.
Their Scholar Xiao, what's going on?
After suddenly writing a lot of mathematical formulas that made people confused, he started laughing, and he was still laughing so happily?
But Liu Bin roughly understood the meaning of Xiao Yi's smile, and stepped forward and asked, "Xiao Yi, do you... have any new ideas?"
Xiao Yi nodded, and suddenly noticed from the corner of his eye that he had not finished writing the last step of the third solution to this problem, so he quickly added the last key step below.
Putting down the pen, he then said to Liu Bin, "Yes, Professor, the screening method, I think I can do it to an even more extreme level."
He said with some embarrassment, "Well... I may have to find a place to write down these thoughts for the time being."
Liu Bin shook his head helplessly.
Didn't I just quibble and say, "I'm in class right now", but now I'm fine, and I'm planning to leave before the class is over.
But what else could he say?
"Go ahead and accomplish more important things."
"Thank you Professor!"
Xiao Yi's face lit up with joy, and he quickly returned to his seat, picked up his books and bag, and left the classroom directly in full view of everyone.
From the perspective of the students, Xiao Yi was just like handing in the paper early in the original exam and finishing get out of class earlier than them on the way to class.
Only Liu Bin knows.
This mathematical genius is going to fight for a greater cause in mathematics again!