Chapter 86: Understand?

Eh~

A bit difficult.

Su Kewei soon finds that the initial proof method doesn't work. After a pause, he exhales, pulls out a draft paper again, and plans to build an upper bound estimate for prime number intervals.

10 minutes passed.

Su Kewei's forehead begins to show fine sweat, and five or six discarded papers have already been piled up.

Xu Qingzhou sees that this senior probably needs some time, takes out his phone, and discusses with Guo Ziyang about buying tickets back for National Day.

In the small group, Guo Ziyang is sending photos of him and Ding Jiahui. The two are shopping at school together, and Guo Ziyang is like a follower in the snack street.

Ding Jiahui: "Yaoyao, when will you come to see us (pitiful)."

Song Yao: "Saturday."

10 minutes passed, the blackboard on the left had long been filled with dense formulas. Gu Zhizhong turned to look at Xu Qingzhou and the others and asked: "Do you understand everything here?"

Read! {

"Cramér's conjecture?" Xu Qingzhou took the blackboard eraser very sensibly and erased the writing on the blackboard just now.

In the Q&A room, many people had already raised their heads, looking at the people on the stage with some surprise. They vaguely heard something about a conjecture just now.

"I encountered a difficult problem and thought about coming to ask." Xu Qingzhou handed over the draft paper directly.

"Understand." Xu Qingzhou and Su Kewei both nodded. Meng Bin thought for a while and also nodded.

Then construct an expression about p, so that when q is the smallest prime number greater than p, there is qp≤f(p).

He said with some surprise: "Senior, have you solved it?"

Gu Zhizhong nodded slightly. Seeing Xu Qingzhou finish erasing the blackboard, he asked curiously: "How far have you solved it?"

Gu Zhizhong stood up, took the blackboard eraser: "This question is indeed very difficult, or rather, it involves a famous mathematical conjecture—Cramér's conjecture."

About 5 minutes passed, Gu Zhizhong looked up at Xu Qingzhou and frowned, asking: "Where did you see this question?"

Gu Zhizhong took the draft paper, put on his glasses, and started to study it. On the side, Meng Bin silently took Gu Zhizhong's teacup and walked toward the hot water room.

"Junior, Junior?"

"Well, this is a way." Gu Zhizhong nodded appreciatively.

When Xu Qingzhou and the three went in, Gu Zhizhong happened to be free and was a bit surprised, "Oh, why did you three come together?"

20 minutes passed.

Su Kewei nodded and said: "I will take you there."

"But there are also problems. Even if such a counterexample is found, it can only prove that the conjecture is not established in this specific situation, but it cannot prove that it is not established in the entire range." Xu Qingzhou sighed, a bit helpless.

Gu Zhizhong continued to write a lot of formulas and said slowly: "Consider the average interval between adjacent prime numbers. For large numbers x, the average interval of adjacent prime numbers is roughly logx."

"The Prime Number Theorem shows that when x tends to infinity, the number of prime numbers less than or equal to x, π(x), is approximately equal to \frac{x}{\logx}logxx."

"Okay." Xu Qingzhou is helpless.

Xu Qingzhou:?

Someone greeted him in a low voice. Xu Qingzhou replied: "I saw it in that 'Higher Mathematics Theory' by Sophia Dawson."

In the group, everyone is planning the activities for this Saturday.

The Q&A room is very quiet, with only the rustling sound of chalk writing on the blackboard.

"This is a book read at the graduate stage." Gu Zhizhong smiled gently and looked at Su Kewei, saying: "You have read it too, right?" Su Kewei nodded embarrassedly. Not only had he read it, but he even bragged that he could solve it.

!.

Seeming to worry that Xu Qingzhou and the others couldn't keep up, he added: "This is derived from the asymptotic expression of the Prime Number Theorem."

Ding Jiahui: "Yaoyao, you remind me of an idiom—the husband sings and the wife follows."

Meng Bin also came back after fetching water at this time.

Xu Qingzhou, who was surfing the group, heard someone calling him, put down his phone, and saw Su Kewei standing beside him.

Professor Gu used a novel method, first introducing the Prime Number Theorem and the prime counting function, while constructing an upper bound function, a complex function f(p) based on the function properties of this function z, which gives a certain measure of the "density" of prime numbers less than p.

Xu Qingzhou took out a pile of draft paper, "I used several methods, but none of them worked. Later, I tried to find a counterexample that violated the conjecture, such as finding a pair of adjacent prime numbers (p, q)(p, q), such that qp>3×(logp)^1.5."

As everyone knows, as long as it is tainted with the word "conjecture", it will not be simple.

"Yes, this question is still a long way from the difficulty of the real Cramér's conjecture, but the difficulty is not small."

"I basically understand this too." Xu Qingzhou nodded.

Song Yao: "I am in class, you and Xu Qingzhou discuss the arrangements."

g(p)=3×(logp)^1.5+(p)pi+1pi≤g(pi)(p, q)=(10007, 10009), the interval is q-p=10009-10007=23×(log10007)^1.5≈3×(9.2103)^1.5≈3×34.406≈103.22.

"Senior Su Kewei, Senior Meng Bin."

This time, he simply wrote down the question, just in time for everyone to brainstorm together.

Gu Zhizhong picked up the chalk while talking about his ideas and wrote on the blackboard.

Q&A room.

Ding Jiahui: "Yaoyao, you are so cold (weak)."

It is just a large classroom. There are many seniors studying on their own below. Most of them are preparing for the postgraduate entrance examination. If they don't understand, they can ask directly. Every day there will be professors who don't agree here.

"Which... not yet." Su Kewei's old face turned red. Noticing Xu Qingzhou's suspicious eyes, he hurriedly said: "I study probability theory and mathematical statistics, well, that's it. I am not good at theoretical mathematics. You still need to find Professor Gu to take a look."

"I'll go too!" Meng Bin stopped writing, wanting to know how to solve the question that even Senior Su was stumped by.

He was a bit emotional. Line by line, he is worthy of being an old professor who has been immersed in the field of mathematics for decades. The thing that troubled him for two days only had ideas after thinking for a few minutes.

Su Kewei also nodded. Although these steps were obscure, he could keep up. He looked at the junior beside him with admiration. He was only a freshman and had already started studying questions of this difficulty.

Xu Qingzhou and Su Kewei both nodded. It was just a bit difficult for Meng Bin. He was already a bit confused when it came to the upper bound function, let alone now.

At this time, the seniors below were also looking blank.

Understand? Understand what?

(End of chapter) ===== CHAPTER 87 =====