The top student must be diligent.

Chapter 128: There is no Landau-Siegel Zero

"We relate the lower bound of L(1,χ) to the distribution of the zeros 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）|is less than 1/2,|（ss0）|】

New mathematical formulas constantly appear on the blackboard.

Xiao Yi's hands never stopped, gradually showing the thoughts in his mind.

Zhang Yitang, who was standing next to him, hugged his arm with one hand and pinched his chin with the other hand. He looked at what Xiao Yi had written and fell into deep thought.

Xiao Yi continued: "Assume ψ∈Ψ. Recall that the function equation of L(s, ψ) was given earlier, in which it was stated that θ is equal to ψ."

"And in the upper half plane, since Z(s, Ψ) does not vanish, there exists an analytic function Y(s, Ψ) such that..."

【Y (s, ψ)2 = Z(s, ψ)1.】

"If we apply Xiao's expansion here, and then use the new polynomial to pick out some key information, we will be able to get..."

"Ok?"

When writing this, Xiao Yi was suddenly startled.

"what happened?"

Seeing Xiao Yi suddenly stunned, Zhang Yitang asked in confusion.

Aren’t we reaching the crucial point now?

What else is there to think about?

However, Xiao Yi didn't say much, but just continued writing on the blackboard.

It’s just that... he wrote on another blank space on the blackboard, and didn’t follow the same line of thought as before.

And what he writes now seems to be a little different from before?
Zhang Yitang was slightly stunned, squinting his eyes and carefully looking at what Xiao Yi wrote, then he said in surprise: "Is this... a round method?"

The circle method, also known as 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 function by dividing the circle into small arcs and long arcs, and then restricting the behavior on the small arcs.

After so many years of development, this method can now be applied to the field of complex analysis.

Zhang Yitang had used this method in his research before, but he had never found a suitable insertion point, so he had to give up in the end.

Perhaps it was because the Yuan method was not sufficiently compatible with the issues he was studying.

But why did Xiao Yi suddenly use it again?

Zhang Yitang wanted to tell Xiao Yi about his previous experience, but seeing Xiao Yi's serious expression, he finally calmed down and continued reading.

While Xiao Yi was writing, he would pause and think about what he had written.

In this way, although his speed was a bit slow, he wrote more and more, gradually filling up the blank space on the blackboard.

However, Zhang Yitang's brows gradually furrowed.

Because he never understood the purpose of what Xiao Yi wrote.

It had never been connected to their problem, and he couldn't see any way to connect it yet.

He temporarily withdrew his thoughts, lowered his head, and looked at the time. It was already past 9 o'clock in the evening.

After driving all day today, he was already quite tired and planned to go back early to rest.

Anyway, the purpose of tonight is to let Xiao Yi know what progress they have made now, and this purpose has already been achieved.

So he said, "Xiao Yi, why don't we stop here today? If you want to use the circle method now, I think..."

Just when he said half of this, the rest of the words got stuck in his throat and he couldn't say it.

Xiao Yi's writing speed suddenly increased and he quickly wrote a line of formulas on the blackboard.

【∑ψ∈Ψ(|X3(P 2，ψ)|+∫|x3(x，ψ)|xdx)2P^2L……】

"This is……"

Zhang Yitang was shocked. What happened?

He just lowered his head for a while, so why did Xiao Yi suddenly bring Yuan Fa into their problem?
He turned back and rethought Xiao Yi's steps, and after a moment, he finally opened his eyes.

"Is it Xiao's development again?"

"It turns out that they import the theta function into the polynomial, and then use Xiao's expansion to reverse the original function, and then introduce it into..."

"Unbelievable, how on earth can this step...?"

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 never have imagined.

The Yuan method was successfully applied to their problems, and more importantly, Xiao Yi also combined the Yuan method with the Xiao family, forming a certain connection between the two.

The result was that all the problems they had encountered before suddenly became clear!

The value of what Xiao has revealed is still rising!
Looking at Xiao Yi's mind, Zhang Yitang really wanted to know how these thoughts appeared in his mind.

For a moment, this 60-year-old mathematician felt what Terence Tao and 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 today, when Xiao Yi came here, he actually found a way to break the deadlock.

It is obvious that the time Xiao Yi can use to think is only in the car and a few hours at night!
Finally, Xiao Yi stopped after completing another step of deduction.

Holding his chin, he pondered for a moment, looking at all the processes on the blackboard, and finally said, "We are almost there, and the big problems have been solved."

Then he turned to look at Zhang Yitang and asked, "By the way, Professor Zhang, what did you say just now?"

"I..." Zhang Yitang came back to his senses and opened his mouth. Uh...

He said 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 swept away all the tiredness and is back to being himself.

Just as Xiao Yi said just now, when he carried the method developed by Xiao into the research, it was as if the Zen barrier was broken, and the most critical problem had been solved.

He now saw the dawn of a complete proof of the Landau-Siegel zero-point conjecture.

The dawn light brought back some of his spirits.

In the following proof, all they need to solve are some remaining small problems.

Finally, he smiled and said, "I think it won't be long before we can go to the celebration banquet."

Xiao Yi also smiled and nodded: "That's true."

"But what I didn't expect was that there was such interaction between the Yuan Method and Xiao's development. Perhaps, there are more possibilities between them..."

Turning his gaze back to the blackboard, his eyes flickered with a faint gleam of enlightenment and he seemed to be thinking again.

Zhang Yitang stopped talking and did not disturb Xiao Yi anymore.

At this moment, he had already deeply realized that he should never influence Xiao Yi when he was thinking.

Thinking back to the time when he was talking to Xiao Yi just now, it must have been the time when the latter was thinking about the crucial point. If his words had affected others, he probably would not have been in such an excited mood as he is now.

As for what Xiao Yi said, there are more possibilities of combining Xiao's expansion with the Yuan method...

He also raised his head and took a look at Xiao Yi's previous argumentation process.

It seems that there is indeed quite a bit to be discovered in the process.

The only problem was that he couldn't figure out from which angle to start digging.

Forget it, let’s not think about it anymore. Anyway, he just wants to solve the Landau-Siegel zero-point conjecture.

However, Xiao Yi did not think about it for too long. He soon shook his head and turned to Zhang Yitang and said, "Okay, Professor Zhang, do you have anything else to research?"

"Me? I don't have any more." Zhang Yitang waved his hands repeatedly: "The biggest problem has been solved, and the remaining small problems... won't take too much effort."

"That's true." Xiao Yi said with a smile, "Then let's stop here today. You must be tired after driving all day. You should rest early. In the remaining few days, we should try to solve the last few minor problems. Maybe we can finish the paper before returning home."

Zhang Yitang laughed: "That's right!"

Then he patted Xiao Yi's arm and said, "It's my blessing to have you."

Xiao Yi waved his hand modestly: "It's mainly because you, Professor Zhang, have produced enough results before, otherwise it wouldn't have been solved so quickly."

"Well, there's nothing much to say about this matter."

Zhang Yitang did not argue too much about this matter.

Anyway, he was very clear that Xiao Yi's achievements saved him many years.

……

After leaving Zhang Yitang's office, they returned to their respective residences.

After returning to the temporary dormitory, Xiao Yi couldn't help but sigh at the environment which was completely different from the previous dormitory.

Can this be considered as a period of traveling life?

It's just that he doesn't have to spend any money at all.

I also made a lot of money...

Sitting at the desk, he took out the draft paper and continued to write content related to the Circle Method on it.

For him, today's learning tasks have not been completed yet, and the remaining time will need to be used for research.

He wanted to further explore the relationship between the circle method and Xiao's expansion.

Perhaps a newer technology can be discovered.

also……

"The circle method is the key method used to solve the Goldbach conjecture."

"If it can be put to use..."

His brows twitched slightly.

After solving the twin prime conjecture, what should be the next mathematical goal?
He thought about many things, such as the n^2+1 prime number conjecture that was already under study, or another Legendre conjecture.

These two problems, like the twin prime conjecture, are all problems related to prime numbers, but now, because of his thinking about the circle method, his attention has gradually turned to the Goldbach conjecture.

In the past, many breakthroughs related to the Goldbach conjecture were inseparable from the shadow of the circle method.

Of course, whether to study the Goldbach conjecture specifically requires some research on the circle method.

See if you can get some results.

Time passed quietly.

……

From then on, Xiao Yi's life in Santa Barbara basically consisted of research.

After a few days, they completely solved all the remaining small problems of the Landau-Siegel zero-point conjecture.

This also means that this conjecture has been completely proven by them!
The final result was not beyond their expectations, that is, there is no Landau-Siegel zero!
(End of this chapter)