Chapter 86 Understand? Uh ~ It's a little difficult. Su Kewei soon discovered that the original proof method did not work. After a pause, he exhaled, took out a new piece of manuscript paper, and planned to construct an upper bound estimate for the prime number interval. 10 minutes passed. Fine beads of sweat began to appear on Su Kewei's forehead, and there were already five or six pieces of discarded paper piled up. Seeing that this senior might take some time, Xu Qingzhou took out his cell phone and discussed with Guo Ziyang about buying tickets to go back home during the National Day holiday. In the small group, Guo Ziyang was posting photos of him and Ding Jiahui, the two of them walking around the school and going to the snack street together. Guo Ziyang is 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. If you have any arrangements, please talk to Xu Qingzhou."
Ding Jiahui: "Yaoyao, your words remind 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 chatting in the group chat, heard someone calling him. He put down his phone and saw Su Kewei standing next to him. He said with some surprise: "Senior, did you figure it out?" "Which one? Not at the moment." Su Kewei's face turned red. Noticing Xu Qingzhou's suspicious look, he quickly said, "I studied probability theory and mathematical statistics. That's it. I'm not good at theoretical mathematics. You still need to talk to 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, wanting to know how to solve the problem that even Senior Brother Su found difficult. Question and answer room. It's just a big classroom. There are many seniors studying there. Most of them are preparing for postgraduate entrance exams. If you don't understand something, you can ask directly. There will be professors here who disagree with you every day. "Senior Su Kewei, Senior Meng Bin." An acquaintance greeted me quietly. When Xu Qingzhou and the other two entered, Gu Zhizhong was free and was a little surprised. "Oh, why are you three here together?" "I encountered a difficult problem and thought of coming over to ask." Xu Qingzhou handed over the manuscript paper directly. Gu Zhizhong took the manuscript, put on his glasses and began to study. At the side, Meng Bin silently took Gu Zhizhong's teacup 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 Sophian Dawson's book "Advanced Mathematical Theory"." "This is a book for graduate students." Gu Zhizhong smiled gently, looked at Su Kewei and said, "You have read it too, right?" Su Kewei nodded a little embarrassedly. Not only had he read it, he even boasted that he could solve it. Gu Zhizhong stood up, holding the blackboard eraser: "This question is indeed difficult, or rather, it involves a famous mathematical conjecture - Cramer's conjecture." "Cramer's conjecture?" Xu Qingzhou took the eraser tactfully and wiped off the writing on the blackboard. "Yes, this question is still a long way from being as difficult as the real Cramer conjecture, but it is still difficult." Gu Zhizhong nodded slightly. When Xu Qingzhou finished wiping the blackboard, he asked curiously, "How far have you solved it?" In the Q&A room, many people have raised their heads and are looking at the person on the stage in surprise. They had just vaguely heard some speculations or something. As we all know, anything involving the word "guess" will not be simple. Xu Qingzhou flipped through a pile of manuscripts. "I tried 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)^." "Well, that's one way." Gu Zhizhong nodded appreciatively. "But there is a problem. Even if we find such a counterexample, 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 helpless. Meng Bin also came back after fetching water at this time. "The prime number theorem states that as x approaches infinity, the number of prime numbers π(x) less than or equal to x is approximately equal to \frac{x}{\log x}logxx." As Gu Zhizhong was talking about his ideas, he picked up a piece of chalk and wrote quickly on the blackboard. This time, he simply wrote down all the questions, allowing everyone to think divergently. The Q&A room was very quiet, with only the rustling sound of chalk writing on the blackboard. g(p)=3×(logp)^(p) Pi+1Pi≤g(Pi) (P, Q) = (10007, 10009), the interval is Q - P = 10009 - 10007 = 2 3×(log10007)^≈3×()^≈3×≈。 Ten minutes later, the blackboard on the left was already filled with densely packed formulas. Gu Zhizhong turned to look at Xu Qingzhou and the others and asked, "Do you understand everything here?" "I understand." Xu Qingzhou and Su Kewei nodded. Meng Bin thought about it and nodded as well. Gu Zhizhong continued to write a bunch of formulas and said slowly: "Consider the average interval between adjacent prime numbers. For large numbers x, the average interval between adjacent prime numbers is roughly log x." As if worried that Xu Qingzhou and the others couldn't keep up, he added: "This is derived from the asymptotic expression of the prime number theorem." "I basically understand this." Xu Qingzhou nodded. The method used by Professor Gu is very novel. He first introduced the prime number theorem and the prime number counting function, and at the same time constructed an upper bound function, a complex function f(p) based on the functional properties of this function ζ, which gives a certain measure of the "density" of prime numbers less than p. Next, construct an expression for p such that when q is the smallest prime number greater than p, qp≤f(p). He was somewhat impressed, thinking one line at a time, as if he was an old professor who had been immersed in the field of mathematics for decades. He had an idea for something that had troubled him for two days after just a few minutes of thinking. Su Kewei also nodded. Although the steps were obscure, he could still keep up. He was impressed by the junior student next to him. He was only a freshman, but he had already started studying questions of such difficulty. Xu Qingzhou and Su Kewei both nodded, but they felt a little embarrassed for Meng Bin. He was already a little confused when he was working on the upper bound function, let alone now. At this moment, the seniors in the audience also looked bewildered.
Understand? Understand what? (End of this chapter)