The top student must be diligent

All the manuscripts he used in the research process are above.

Xiao Yi took the draft paper and started to read it from the beginning.

The front of this stack of draft paper is mainly the results made by Zhang Yitang during this period.

"Well... you have proved that the inequality V1j(y)T^c follows the Polya-Vinogradov inequality and the partial summation."

"Oh? You have also given the relationship between the lower bound of L(1, χ) and the non-vanishing central value of this key automorphic L function."

"And..."

Soon, Xiao Yi nodded: "Now it is indeed very close to the final answer."

"Haha, it's mainly because the Xiao expansion method is useful enough." Zhang Yitang said with a smile while driving.

Otherwise, when he saw Xiao Yi's paper at that time, he would not be so surprised.

"Hmm..." Xiao Yi nodded slightly and continued to turn to the next draft paper: "Then, the remaining problem is..."

"It is necessary to solve the zero point problem of L(s, ψ) in Ω, mainly to prove that the gap between any different zero points of A(s, ψ) in Ω is greater than α(1c′αL)."

"And use this to prove the most basic proposition. When θ(mod k) is the primitive feature, the function equation of L(s, θ) is L(s, θ)=Z(s, θ)L(1s, θ)."

"These questions..."

Looking at the description of these problems, Xiao Yi began to think.

Zhang Yitang stopped talking and drove quietly.

The only sounds around were the engine sound of the car, the wind sound outside the car window, and the sound of waves from the sea.

...

The last few problems were also not easy to solve.

However, for Xiao Yi, even though it was difficult to come up with a direct solution, he still found an angle that could achieve a breakthrough.

That afternoon, he arrived at the University of California, Santa Barbara, and under Zhang Yitang's arrangement, Xiao Yi once again moved into the student dormitory of the University of California, Santa Barbara.

After a simple dinner, he and Zhang Yitang came to the latter's office to discuss the Landau-Siegel zero conjecture.

Chapter 128 There is no Landau-Siegel zero

Chapter 128 There is no Landau-Siegel zero

"We link the lower bound of L(1, χ) with the zero 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)| is less than 1/2, |(ss0)|]

New mathematical formulas kept appearing on the blackboard.

Xiao Yi's hands did not stop for a moment, gradually showing the ideas in his mind.

Zhang Yitang, who was standing next to him, was holding his arm with one hand and holding his chin with the other hand, looking at what Xiao Yi had written and lost in thought.

Xiao Yi continued: "Assuming ψ∈Ψ, recall that after the function equation of L(s,ψ) was given earlier, it was stated that θ equals ψ."

"And on the upper half plane, since Z(s,ψ) does not disappear, there exists 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 from the new polynomial, we can get..."

"Hmm?"

When writing here, Xiao Yi was suddenly startled.

"What's wrong?"

Seeing Xiao Yi suddenly stunned, Zhang Yitang wondered.

Isn't it the key point now?

What else needs to be thought about?

However, Xiao Yi did not say much, but continued to write on the blackboard.

It's just that... he wrote on another blank space on the blackboard, not along the same line of thought.

And what he wrote now seems to be a little different from before?

Zhang Yitang was slightly stunned, squinted his eyes and looked carefully at what Xiao Yi wrote, and then said in surprise: "Is this... the circle 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 residual of a function n~F(n) that is approximately zero. The method is to divide the circle into small arcs and long arcs, and then restrict 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 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.

Maybe it was because the circle method was not compatible with the problem he was studying.

But why did Xiao Yi suddenly use it again?

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

Xiao Yi would pause while writing, looking at what he had written and thinking.

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 frowned.

Because he still didn't understand the purpose of what Xiao Yi wrote.

It has never been connected to their problem, and he can't see any way to connect it for the time being.

He temporarily pulled out 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 and rest early.

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

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

Just halfway through, the rest of the words got stuck in his throat and couldn't be said.

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, how did Xiao Yi suddenly involve the circle method in their problem?

He turned back and rethought Xiao Yi's steps, until a moment later, he finally widened his eyes.

"Is it Xiao's expansion again?"

"It's actually importing the θ function into the polynomial, and then inferring the original function based on Xiao's expansion, and then introducing it into..."

"Incredible, how is this step...?"

Although he had no idea what the other party's thinking was like, Zhang Yitang finally understood what the things Xiao Yi wrote were for.

In silence, he completed the inference that he could not imagine at all.

The circle method was successfully applied to their problem, 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 were suddenly clear!

The value of Xiao's expansion is still rising!

Looking at Xiao Yi's mind, Zhang Yitang really wanted to know how these ideas appeared in his mind.

For a moment, this 60-year-old mathematician felt what Terence Tao and his team felt on the first day.

You know, these problems have been bothering him for a long time, but he has never found a solution. Today, Xiao Yi came and found a way to break the deadlock.

Xiao Yi obviously only had time to think in the car and a few hours at night!

Finally, Xiao Yi stopped after completing another step of derivation.

He pinched his chin and pondered, looking at all the processes on the blackboard, and finally said: "It's almost here, and the big problems are solved."

Then he turned his head 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 to rest?

Just kidding, if he was a little tired just now, then now he has swept away the fatigue and become himself again.

Just as Xiao Yi said just now, when he carried Xiao's circle method into the research, it was as if the gate of Zen was broken, and the most critical problem had been solved.

He now saw the dawn of completely proving the Landau-Siegel zero-point conjecture.

This dawn of light made him feel a little refreshed.

In the next proof, they only need to solve some remaining small problems.