Chapter 86 Understand?
Eh~
It’s a little difficult.
Su Kewei soon discovered that the original proof method did not work. After a pause, he exhaled and took out another piece of paper, intending to construct an upper bound estimate of the interval between prime numbers.
10 minutes passed.
Beads of sweat began to appear on Su Kewei's forehead, and five or six discarded papers had been piled up.
Xu Qingzhou saw that the senior was expected to take a while, so he took out his mobile phone and discussed with Guo Ziyang about buying tickets to go back during the National Day.
In the small group, Guo Ziyang was sending photos of him and Ding Jiahui. The two of them visited school and went to the snack street together
Guo Ziyang was like a follower.
Ding Jiahui: "Yaoyao, when will you come to see us (poor)."
Song Yao: "Saturday."
Ding Jiahui: "Yaoyao, you are so cold (weak)."
Song Yao: "I'm in class, tell Xu Qingzhou if you have any arrangements."
Ding Jiahui: "Yao Yao, this reminds me of an idiom - the husband sings and the wife follows."
Xu Qingzhou:?
In the group, everyone is planning activities for this Saturday.
20 minutes passed.
"Junior, junior?"
Xu Qingzhou, who was in the water group, heard someone calling him, put down his mobile phone, and saw Su Kewei standing next to him.
He said with some surprise: "Senior, have you solved it?"
"Which. Not yet?" Su Kewei's old face turned red. Sensing Xu Qingzhou's suspicious eyes, he quickly said: "I studied probability theory and mathematical statistics. Well, that's it. I'm not good at theoretical mathematics. You still need to Check with Professor Gu.”
"Okay." Xu Qingzhou was helpless.
Su Kewei nodded and said, "I'll take you there."
"I'll go too!" Meng Bin stopped writing and wanted to know how to solve the problem that even Senior Brother Su had trouble with.
Q&A room.
It's just a big classroom. There are many seniors studying by themselves below. Most of them are taking postgraduate entrance examinations. You can ask directly if there are any. Every day there are professors here who don't agree with me.
"Senior Su Kewei, Senior Meng Bin."
An acquaintance said hello in a low voice.
When the three of Xu Qingzhou entered, Gu Zhizhong happened to be free and was a little surprised, "Oh, why did you three come here together?"
"I encountered a problem and wanted to come over and ask." Xu Qingzhou handed over the manuscript paper directly.
Gu Zhizhong took the manuscript paper, put on his glasses and began to study. On the side, Meng Bin silently took Gu Zhizhong's tea cup and walked towards the hot water room.
About 5 minutes later, Gu Zhizhong looked up at Xu Qingzhou, frowned and asked, "Where did you see this question?"
Xu Qingzhou replied: "I saw it in Sophie Dawson's "Advanced Mathematics Theory"."
"This is a book I read when I was a graduate student." Gu Zhizhong smiled softly, looked at Su Kewei, and said, "You have read it too, right?" Su Kewei nodded a little embarrassedly. Not only had he read it, but he even boasted that he could Solve it.
Gu Zhizhong stood up and took the blackboard eraser: "This question is indeed difficult, or in other words, it involves a famous mathematical conjecture - Kramer's conjecture."
"Krammel's conjecture?" Xu Qingzhou took the blackboard eraser very wisely and erased the writing on the blackboard.
"Yes, this question is still far from the difficulty of Kramer's conjecture, but it is not easy."
Gu Zhizhong nodded slightly. Seeing Xu Qingzhou finish wiping the blackboard, he asked curiously: "How far have you reached?"
In the question and answer room, many people had raised their heads and looked at the people on the stage with some surprise. They had just vaguely heard some speculation.
As we all know, anything with the word "guess" will not be simple.
Xu Qingzhou pulled out a pile of manuscript paper, "I tried several methods, but none of them worked. Later, I tried to find counterexamples that violated the conjecture, such as finding a pair of adjacent prime numbers (p, q) (p, q), such that qp >3×(logp)^1.5.
"Well, this is a way." Gu Zhizhong nodded appreciatively.
"But there is a problem. Even if such a counterexample is found, it can only prove that the conjecture is not true in that specific case, but it cannot prove that it is not true in the entire range." Xu Qingzhou sighed, feeling a little helpless.
Meng Bin also came back from fetching water at this time.
"The prime number theorem shows that when x approaches infinity, the number of prime numbers π(x) less than or equal to x is approximately equal to \frac{x}{\logx}logxx."
While Gu Zhizhong was talking about his ideas, he picked up the chalk and wrote on the blackboard.
This time, he simply wrote down all the questions, which just allowed everyone to think together.
The question and answer room was quiet, except for the rustling sound of chalk writing on the blackboard.
g(p)=3×(logp)^1.5+(p)
Pi+1Pi≤g(Pi)
(P, Q)=(10007, 10009), the interval is Q-P=10009- 10007=2
3×(log10007)^1.5≈3×(9.2103)^1.5≈3×34.406≈103.22.
Ten minutes later, the blackboard on the left was already filled with dense formulas. Gu Zhizhong turned to look at Xu Qingzhou and the others and asked, "Do you understand everyone here?"
"Understood." Xu Qingzhou and Su Kewei nodded. Meng Bin thought for a while and nodded.
Gu Zhizhong continued to write a lot of formulas and said slowly: "Consider the average distance between adjacent prime numbers. For a large number x, the average distance between adjacent prime numbers is roughly logx."
As if worried that Xu Qingzhou and the others would not be able to keep up, he added: "This is derived from the asymptotic expression of the prime number theorem."
"We basically understand here." Xu Qingzhou nodded.
The method used by Professor Gu is very novel. He first introduces the prime number theorem and the prime number counting function, and at the same time constructs an upper bound function. Based on the functional properties of this function ζ, the complex function f(p) gives the "density of prime numbers less than p". "Some kind of measure.
Then construct an expression about p such that when q is the smallest prime number greater than p, qp≤f(p).
He actually sighed a little, one line after another. It was indeed an old professor who had been in the field of mathematics for decades. He had troubled him for two days, but he already had an idea after just thinking for a few minutes.
Su Kewei also nodded. Although these steps were obscure, he could still follow them. He was very impressed with the junior student next to him. He was only a freshman and he had already started studying such difficult topics.
Xu Qingzhou and Su Kewei both nodded, but Meng Bin was a little embarrassed. He was already a little confused when it came to the upper bound function, let alone now.
At this time, all the senior students in the audience were equally confused.
Understand? What do you know?
(End of this chapter)