Mathematician Eichler once said that there are only five basic operations in mathematics, addition, subtraction, multiplication, division, and modular form.
Maybe this is a very personal argument, but it does show that modular forms are everywhere in today's more profound mathematics.
The modular form is an analytic function, and under a group operation on the model group, this function will become a certain type of functional equation, and the value calculated through the function will also show a certain growth trend.
All in all, this is a mathematical tool that is widely used, including the Taniyama Yutaka-Shimura Goro conjecture in the proof of Fermat's last theorem, and modular formal theory is also used.
Research on the connection between modular formalism and group theory is also widespread in the mathematical community.
At this moment, Lin Xiao saw such an application in the proof process from Professor Selberg.
"Yeah, why didn't I think of that?"
While he was absorbing the knowledge contained in these manuscripts as if he was receiving an enlightenment, he was also sighing at the inspiration the knowledge contained therein brought to him.
It is worthy of Professor Atler Selberg who once proved the prime number theorem. Such application of knowledge is worthy of the masterpiece of the master.
"After studying group theory and Lie groups for so long, I can't imagine such a change. This will match my previous steps..."
Lin Xiao's eyes became brighter and brighter.
Then he quickly picked up a pen and scratch paper and began the next performance.
These manuscripts did not tell him the answer. After all, Professor Atler Selberg had not succeeded, so he needed to follow the ideas conveyed in the manuscripts to complete the next work.
The next work is a big project. He needs to combine group theory with his previous research, and then integrate modular formal theory into it to complete a new mathematical method that is more applicable.
And through this new method, he had a hunch that he would be able to achieve his ultimate goal.
However, he had to give it a try before he realized that after all, sometimes inspiration can become an illusion.
Even Professor Selberg only used this method, but it was not successful in the end.
Of course, having a direction is the most important thing for mathematical research.
So Lin Xiao continued his work.
[...F2k(τ)=(2ζ(2k))^(-1)G2k(τ),τ∈H.
…
log|P(ζ1,ζ2,ζ3)|>-μT^4(logT)^24……]
The more Lin Xiao writes, the more he feels like a fountain of thoughts. Although this does not allow him to find the state where his brain development reached more than 4% in the sage state, this state of continuous emergence of inspiration is also commendable.
Even if he encounters some problems in the middle, he can still solve them quickly.
In this way, with the rows of formulas listed, and piles of continuous combinations of English letters, Arabic numerals, and Greek letters, he used the pieces of draft paper, and he threw some aside.
, some were kept by him.
Time passes more and more.
He could already feel that he was closer than ever to the final result.
I don't know how long later, Sun Yu, Ouyang Sheng and others, who had also finished the exam, had also returned with the two team leaders. When they saw Lin Xiao's closed room, they all shook their heads and did not disturb him.
"I don't know if Lin Shen can solve that problem."
"Since everyone calls him Lin God, all you need to do is believe in God."
"In the future I will have to make a memorial tablet for Lin Shen and offer it at home every day. I hope I can be half as awesome as him."
"Are you half awesome? You might want to go to heaven."
Several people laughed and joked, and then continued to review their exam process and answer questions.
However, all in all, they all felt pretty good about this exam and their performance was pretty good. Except for the third and sixth questions, which were both final questions and which they found very difficult, they still had a certain degree of confidence in the other questions.
In this way, they chatted until the afternoon, and even stopped by for a meal. As for Lin Xiao, although they did not eat with them, they still helped Lin Xiao bring food back. In the past, Lin Xiao once
When they entered the research state and couldn't think of eating, they would bring food for Lin Xiao.
Until evening, they began to plan to have a late-night snack outside to make up for the regret of having to eat in the hotel these past few days. By the way, they could also experience the local customs and customs here.
However, just as they were about to go out, the closed door was opened and Lin Xiao walked out.
"Hey, Lin Shen, it's a coincidence that you came out. We were just talking about going out."
"Are you going out to eat?" Lin Xiao asked.
"yes."
"Then let's go." Lin Xiao touched his stomach and said awkwardly: "I'm a little hungry."
"Oh my god, Lin Shen is still hungry!"
"Then if you don't leave quickly, how can you make Lin Shen hungry?"
The group of people immediately started shouting and yelling. They grabbed Lin Xiao and started to set off.
Lin Xiao looked at them like this and couldn't help but shook his head.
From 10:30 in the morning to about 3 o'clock in the morning, he only ate one meal of the Romanian specialty cabbage rolls they brought back and a small plate of cannelloni. It was already 10 o'clock, and he was really good.
hungry.
This was also a rare occasion when he was directly starved out of his immersive state.
Of course, after such a long period of research, he has achieved great results. Although he has not succeeded yet, he is now basically sure that he is on the right direction. As long as he continues to go, success will be visible.
However, whether a person is iron rice or steel, he should first fill his stomach before talking about it.
Afterwards, Lin Xiao left the hotel with them and went out to have supper and enjoy the scenery.
…………
The next two days will be the time for the IMO test paper marking team to mark the test papers. At the same time, each contestant will be organized to go on an excursion.
However, Lin Xiao, who was dedicated to his own problems, did not participate in this event and continued to stay in the hotel doing research.
Finally, on July 12, his research came to an end.
Looking at the long series of movements in front of him, Lin Xiao looked back at the previous process, and finally let out a sigh of relief: "This is it, it's done."
"So, should I call it Lin's group transformation theory or Lin's modular form transformation method?"
"Hey, forget it, don't worry about this for now, let's settle the last problem first."
The final question is the distribution of Mersenne prime numbers.
At this moment, the answer that seemed to be separated by a chasm was already at his fingertips.
