Chapter 307: The proof is out? Xu Qingzhou grinned, took a deep breath to refresh himself, and said, "Okay, Senior Brother, I'll take a shower first before going over." "I'll wait for you in the lobby downstairs." Su Kewei was a little worried. He felt that Xu Qingzhou seemed very tired. If he was careless and late, it would inevitably cause trouble for others. Xu Qingzhou nodded, took clean clothes and went to take a shower. Soon the sound of running water could be heard in the bathroom. There was a breakfast plate on the dining table in the room, which should have been the breakfast that the waiter had ordered directly. One corner of the desk was piled with discarded manuscripts, and the table was also covered with manuscript paper. Outside the window, the weather was clear and sunny, and the bright sunshine came in through the window and scattered on a corner. The breeze blew the curtains and lifted up a corner of the manuscript paper that was pressed by the cup on the desk. If someone looks carefully, they can find the following content: 【在γ(T)上|F(s)|=|xs+1s(s+1)(ζ′(s)ζ(s)|t|2|t|2=Axc+1|t|22 It is easy to prove that ∫(Re(s)=c)+γ(T)F(s)ds=0. 】 【让 h1<h2<hk取遍大于k的素数，则根据Lagrange定理可知素数p>k时总有νp≤k<p。而当p≤k时 Q(0)=h1h2…，hk}便是一个恰有k个元素的可行整数】
[That is: for all natural numbers k, there are infinite prime number pairs (p, p + 2k), and the Polignac conjecture is proven. ] That’s right, after three days of hard work, Xu Qingzhou proved the Polignac conjecture. Although he was temporarily out of the control of School Beauty Song, he still didn't reach the point where he would stay up all night. In the past few days, he went to bed at 2 o'clock and got up at 6 o'clock in the morning. I stayed up all night last night. A world-class achievement is right in front of me and I can't sleep at all. It took him one night to go through the entire proof process three times. Although some parts can be simplified, the logic and proof are fine. We will check it a dozen times to find bugs and then publish the results. 9 points. After taking a shower, Xu Qingzhou finally felt much more sober. He changed his clothes, took the USB drive and files needed for the report, and went downstairs to find Su Kewei. Seeing Xu Qingzhou coming down, Su Kewei put away his notebook, and the two of them left the hotel and headed towards the lecture hall. Su Kewei asked curiously: "Junior brother, is the progress of Polignac's conjecture going smoothly?" "It's done." "Oh, it's okay, after all, it's just one." Su Kewei wanted to comfort this junior brother, but he suddenly stopped and looked at Xu Qingzhou in astonishment: "You said you proved it?!" "It's proven." "You proved the Polignac conjecture?!" "Ah." At 9:50, Xu Qingzhou arrived at the lecture hall, contacted the staff, and prepared to go on stage. There were many familiar faces in the lecture hall, including Professor Qin Yizhen, Chairman Cai Xinyuan, and President of the International Mathematical Union Shigefumi Mori. There were quite a lot of people, most of whom were experts in the field of number theory, scholars who had studied number theory, or people who were just there to join in the fun.
Su Kewei looked dazed and waved to Professor Gu in the crowd. Damn, my junior brother has come up with another world-class conjecture! He wanted to report this matter to Gu Zhizhong, but found that Professor Gu was discussing the Selberg formula problem on arithmetic progression with Academician Chu with the manuscript in hand, so he decided to wait for a while. 4 minutes passed. "Well, then by repeatedly using the property ∑n≤x(n, k)=1f(n)=∑.d|n, d|kμ(d) and appropriately shifting terms, we can get the formula. Old Chu, come down and talk. Let's listen to this kid's report first." "Hehe, okay." Chu Jiangfeng nodded with a smile. When Gu Zhizhong finished speaking, he turned around and found Su Kewei in a daze. "Xiao Su." "Professor." Su Kewei came back to his senses and finally remembered to talk about his junior's amazing deeds. "Xiao Su, you are also very interested in the sieving method. You can discuss it more with your junior brother." "Yes, I remember that, Professor." Gu Zhizhong waved his hand and said, "We'll talk about other things later. Let's listen to the lecture first." Su Kewei wanted to say something but stopped himself. Seeing that Gu Zhizhong's eyes were already looking at Junior Brother Xu on the stage, he sighed helplessly and said, "Well, let Junior Brother Xu say it himself." At 10 o'clock, the report officially began. "There are two main contents of this report. The first is the improvement of the sieve method of the twin prime theorem and the simplification of the calculation of the upper bound M. The second is to introduce to you a new tool that I summarized when studying the Polignac conjecture." Xu Qingzhou stood on the stage, simply greeted the people in the audience, and then got to the point. He first talked about the twin prime theorem. "If P(z) represents the product of all prime numbers whose size does not exceed z, then the previous sieve method can be written as: S = ∑N “当：gi(d)=μhi(d)=μ2(d)∏p|dgi(p)1gi(p)第67就可以化成：S=[1+o(1)]NlogN((R2log3kR)+O(E)” The content of the first half is not new, it just supplements the previous proof process. After speaking for 20 minutes, Xu Qingzhou moved on to the second part. "As I said at the beginning, in studying the Polignac conjecture I created a new tool - the harmonic sieve." There was a commotion in the audience. "Harmonic sieving method? Is it an improved version of a classic sieving method?" "It should be." Everyone started discussing in low voices. The sieve method is one of the most effective tools for finding prime numbers or solving problems related to prime numbers. Common sieve methods include the sieve of Eratosthenes, the interval sieve, etc., or improved versions of these sieve methods. Gu Zhizhong nodded slightly, his eyes curious, wanting to know what kind of screening method this boy had used. "In order to better study the distribution of prime numbers, I used the Selberg sieve as a basis, in which I used solution sets and number sequences to explore the properties of twin primes." Xu Qingzhou went straight to the point and brought out all the formulas.
The lecture hall was filled with the sound of flipping through notebooks. In front, Xu Qingzhou has already started: "Using (4), we get: 1(ΛΛ+Λ′)=1″. Doing the Möbius inversion on both sides, we get: ΛΛ+Λ′=μ1″." "Get the definition of Dirichlet convolution and the definition of derivatives fixed: ∑rd=nΛ(r)Λ(d)+Λ(n)lum_{rd=n}\mu(r)\log^2dag5” This concludes the narrative part of the report. In the audience, many people were amazed and felt that this screening method was perfect. It's time for questions. Apparently, there is a lot of interest in the Harmonic Sieve method. For example, a middle-aged professor stood up and asked: "On page 53 of the PPT, the necessary and sufficient condition for d to be solved is that q and k are relatively prime. We only need to consider the case where q and k are relatively prime. How did you get this?" Xu Qingzhou thought for a moment and said, "Using the method of summation by parts, we only need to process the content on the right side of the equation: ∑qd≤xqd≡h(k)μ(q)log2d=∑q≤x(q,k)=1μ(q)∑d≤xd≡q1h(k)log2d” There are also questions about how to incorporate harmonic number sequences into the sieve method. Xu Qingzhou answered them one by one. The sixth person asked a question, and the microphone went to an old acquaintance. Pang Handong, Gu Zhizhong's old rival. Pang Handong did not raise the question of the sieving method, but asked with a smile: "Student Xu Qingzhou, is this blending sieving method your entire achievement in the past six months?" "It's part of it." Xu Qingzhou answered calmly, realizing that the old man was looking for trouble. "Part of it?" Pang Handong smiled and continued, "Half a year ago, you used the Polignac conjecture as a project, and now many research institutions in the world have received good news. It seems that you haven't heard much news here. We are all very anxious." Xu Qingzhou said calmly, "One of my teachers once said that one must be able to concentrate on one's studies. I think this is right, and I have always taken this as my code of conduct." Pang Handong's face froze. Everyone knew that old man Gu Zhizhong once ridiculed him for not concentrating on studying and instead dabbling in miscellaneous things. Professor Gu also smiled slightly, thinking that this boy really has his own style. Pang Handong suppressed his anger and said with a forced smile, "Number theory, you young people should have some ideas before you reveal your progress. After all, many scholars like me are looking forward to hearing good news from you." As far as he knew, this person had participated in two major projects of the Institute of Physics since he applied for the project. No matter how talented a person was, it was impossible for him to spare the energy to study other things. Didn’t you, Gu Zhizhong, say that one should be steady in doing research? Why are your students so inconsistent? Although it wouldn't be too extreme, it would still be simple to just pull Xu Qingzhou over the fire and roast him. I just want you to tell me that the proof of Polignac's conjecture is not going well. (End of this chapter)