"We relate the lower bound of L(1, χ) to the zero-point distribution of the function L(s, ψ) L(s, ψχ), and then introduce P=exp{L^9}."
"Let Ψ be the set of all primitive characters ψ (mod p) with p ~ P, and let..."
[Ω={s:| (ss0)|less than 1/2, | (ss0)|]
On the blackboard, new mathematical formulas constantly appear.
Xiao Yi's hands kept moving, gradually revealing the thoughts in his mind.
Next to him, Zhang Yitang was holding his arm with one hand and holding his chin with the other hand. He was looking at what Xiao Yi had written and was lost in thought.
Xiao Yi continued, "Assuming ψ∈Ψ, recall that after the function equation of L(s, ψ) was given previously, it has been stated that θ is equal to ψ."
"And on the upper half plane, since Z(s, ψ) will not disappear, there is an analytical function Y (s, Ψ) such that..."
[Y (s, ψ)2 = Z(s, ψ)1.]
"If we apply Xiao's expansion here, and then pick out some key information based on the new polynomial, then we can get..."
"Huh?"
When writing this, Xiao Yi was suddenly startled.
"What's wrong?"
Seeing Xiao Yi suddenly stunned, Zhang Yitang made a confused sound.
Aren't we getting to the key point now?
What else do you need to think about?
However, Xiao Yi didn't say much and just continued writing on the blackboard.
It's just that... he wrote in another blank space on the blackboard, instead of following the same idea as before.
And what he writes now seems a little different from before...?
Zhang Yitang was slightly dazed, squinted his eyes and looked carefully at what Xiao Yi wrote, and then said in surprise, "Is this... round method?"
The circle method, whose full name is also called the Hardy-Littlewood circle method, is one of the most commonly used mathematical techniques in Analytic Number Theory.
Specifically, it is used to calculate the approximately zero residual in n~F(n) of a certain function by dividing the circle into small arcs and long arcs, and then limiting the behavior on the small arcs.
After so many years of development, this method can also be applied to the field of complex analysis.
Zhang Yitang has used this method in research before, but he has never found a suitable insertion point, so he had to give up in the end.
Perhaps it is because the round method is not compatible with the problem he is studying.
But why is Xiao Yi suddenly used again?
Zhang Yitang wanted to talk to Xiao Yi about his previous experience, but looking at Xiao Yi's serious expression, he finally calmed down and continued reading.
Xiao Yi While writing, he would pause and look at what he had written and think about it.
In this way, although the speed was a bit slow, he wrote more and more things, gradually filling up the blank space on the blackboard.
However, Zhang Yitang's brows gradually wrinkled.
Because he never understood the purpose of what Xiao Yi wrote.
There was still no connection to their problems, and he couldn't see any way to connect them for the time being.
He temporarily pulled out of his thoughts, lowered his head, and looked at the time. It was already past 9 p. m.
After driving all day today, he was already quite tired and planned to go back and rest early.
Anyway, the purpose of tonight is just to let Xiao Yi know where they are now, and this purpose has already been achieved.
So he said, "Xiao Yi, how about we stop here today. If you want to use the circle method now, I think..."
As soon as I said this halfway, the rest of the words got stuck in my throat and I couldn't say them out.
Xiao Yi's writing speed suddenly accelerated and he quickly wrote a line of formula on the blackboard.
[∑ψ∈Ψ(|X3(P 2,ψ)|+∫|x3(x,ψ)|xdx)2P^2L...]
"This is..."
Zhang Yitang looked shocked, what happened?
Didn't he lower his head for a while, Xiao Yi why did he suddenly intervene in their problems?
He turned back and rethought the steps of Xiao Yi until after a moment, he finally opened his eyes.
"Is it actually Xiao's expansion again?"
"I actually introduced the θ function into the polynomial, and then reversed the original function based on Xiao's expansion, and then introduced it into..."
"Incredible, what exactly is this step like...?"
Although he had no idea what the other party was thinking, Zhang Yitang finally understood the purpose of what Xiao Yi wrote.
In silence, he completed an inference that he could not imagine at all.
The circle method was successfully applied to their problems, and more importantly, Xiao Yi also combined the circle method with Xiao's expansion, forming a certain connection between the two.
The result of this is that all the problems they encountered before suddenly became clear!
The gold content of Xiao's expansion is still rising!
Looking at Xiao Yi's mind, Zhang Yitang wanted to know how these thoughts appeared in his mind.
For a time, this 60-year-old mathematician realized what Terence Tao and the others felt on the first day.
You know, these problems have been bothering him for a long time, but he has never been able to find a solution. But when Xiao Yi came over today, he actually found a way to solve the problem.
Obviously the time Xiao Yi can use to think is just in the car and a few hours at night!
Finally, Xiao Yi stopped after completing another step of derivation.
Holding his chin, he pondered, looked at all the processes on the blackboard, and finally said, "That's almost it, the big problems have been solved."
Then he turned his head and looked at Zhang Yitang and asked, "By the way, Professor Zhang, what did you just say?"
"I..." Zhang Yitang came back to his senses and opened his mouth.
Uh...
Said that he was a little tired and wanted to go back and rest?
Just kidding, if he was a little tired just now, now he has overcome the fatigue and is himself.
Just as Xiao Yi said just now, when carrying Xiao's expansion's round method into the research, it seems that the Zen barrier has been broken, and the most critical issue has been solved.
He has now seen the dawn of complete proof of Landau-Siegel-Zero Conjecture.
This dawn restored his spirit a little.
In the next proof, all they need to solve are some remaining minor problems.
Finally, he showed a smile and said, "I said, I don't want it for long, we can just go to the celebration banquet."
Xiao Yi also smiled and nodded, "That's true."
"What I didn't expect a little bit is that there can be such interaction between Yuan Fa and Xiao's expansion. Perhaps, there will be more possibilities between them..."
Turning his eyes back to the blackboard, his eyes flashed with a faint light of thought, and he started to think again.
Zhang Yitang stopped talking and did not disturb Xiao Yi again.
At this time, he has deeply realized that he should never influence Xiao Yi when he is thinking.
Recalling the moment when he was talking to Xiao Yi, it must have been the moment when the latter was thinking about the key. If his words had affected others, he probably wouldn't be as excited as he is now.
As for what Xiao Yi said, there are more possibilities to combine Xiao's expansion with the circle method...
He also raised his head and glanced at the arguments in front of Xiao Yi.
It does seem that there are quite a few processes that can be unearthed.
But the problem is that he can't figure out from which angle to start digging.
Forget it, it's better not to think about it. Anyway, he just wants to get Landau-Siegel-Zero Conjecture done.
However, Xiao Yi did not think for too long. He quickly shook his head, turned to Zhang Yitang and said, "Okay, Professor Zhang, do you have anything else to study?"
"Me? I don't have it anymore." Zhang Yitang waved his hands repeatedly. "The biggest problem has been solved, and the remaining small problems... won't take much effort."
"That's true." Xiao Yi said with a smile, "Then let's stop here today. You should be tired after driving for a day, so you should rest early. In the remaining days, try to solve the last few small problems. Maybe we can finish the paper before we return to China."
Zhang Yitang laughed, "That's right!"
Then he patted Xiao Yi on the arm and said, "I am lucky to have you."
Xiao Yi waved his hand modestly. "The main reason is that Professor Zhang has achieved enough results before, otherwise it would not have been solved so quickly."
"Okay, there's nothing to say about this."
Zhang Yitang didn't argue too much on this matter.
Anyway, he knows very well that the results of Xiao Yi have saved him many years of time.
After leaving the office of Zhang Yitang, they also returned to their residences.
After returning to the dormitory where he stayed temporarily, Xiao Yi couldn't help but sigh when he saw the completely different environment from the previous dormitory.
Isn't this a sojourn life?
It just doesn't cost him any money at all.
I also made a lot of money by the way...
Sitting at the desk, he took out the draft paper and continued to write about the circle method on it.
For him, today's study task has not been completed, and the remaining time still needs to be used for research.
He wanted to further explore the relationship between the circle method and Xiao's expansion.
Maybe you can discover a newer technology.
In addition...
"The circle method is the key method used to solve Goldbach Conjecture."
"If it can be used..."
His brows raised slightly.
After solving Twin Prime Conjecture, what should be your next math goal?
He has thought about many things, such as the n^2+1 prime number conjecture that is already under study, or another Legendre's conjecture.
These two questions, like Twin Prime Conjecture, are all about prime numbers, but now, because of his thinking about the circle method, his eyes gradually turned to Goldbach Conjecture.
In the past, many breakthroughs related to Goldbach Conjecture were inseparable from the shadow of Yuan Fa.
Of course, whether to study Goldbach Conjecture specifically requires some research on the circle method.
See if you can get some results.
Time also passed quietly.
Next, Xiao Yi's life in Santa Barbara basically consisted of research.
In a few days, they completely solved all the remaining minor problems of Landau-Siegel-Zero Conjecture.
This also means that this conjecture has been completely proven by them!
And the final result was not beyond their expectations, that is, there is no Landau-Siegel zero point!