Chapter 254

The cross-university project has been open for half a month.

In the past half a month, both Luo Yu's research group and Bao Zi's research group have made good progress.

Especially Bao Zi's side.

Several major breakthroughs have been made in the problem of the mean of the Fourier coefficients of the self-defended L-function with the variable of ternary quadratic type.

On Bao Zi's desk there was a document.

It records in detail a series of progress made by Bao Zi in the process of studying the mean of the Fourier coefficients of the self-supporting L-function.

Gu Lu glanced at it briefly, and then nodded with satisfaction.

In general, the progress of Bao Zi's side is faster than Gu Lu expected.

Gu Lu thought that in about half a month, Bao Zi's progress of 10% would be considered good.

But now it seems that this estimate is still a bit conservative.

In the research on the mean Fourier coefficient in the project, the progress of Bao Zi's side is almost 15%.

If this efficiency continues, it will not take half a year, but only about four months, and it will be completed on the Baozi side.

Gu Lu still didn't know the progress of Luo Yu's research at Jinling University, so he couldn't make a comparison.

But the research progress of Bao Zi's side must not be called slow.

Gu Lu was flipping through the documents on the table while waiting for Bao Zi to come back.

Ten minutes later.

"Teacher, I'm back!"

Before anyone arrives, the sound comes first.

Looking up after hearing the voice, Gu Lu happened to see Bao Zi jumping in, carrying a packing bag in his hand.

There were a few large buns in the bag, presumably for breakfast.

Bao Ziyang raised the bag in his hand, "Teacher, do you eat steamed buns?" ”

Gu Lu smiled and waved his hand, "No, I've already eaten breakfast." ”

Bao Zi pulled a chair and sat next to Gu Lu, a bun was bitten off in half, and he said vaguely while eating, "Teacher, now you can guide me through the problem I encountered." ”

"Are you sure it's now?" Gu Lu pointed to the unfinished bun in Bao Zi's hand.

Bao Zi nodded, "This saves time." ”

"Then you can say it. ”

At the behest of Gu Lu, Bao Zi talked about a difficult problem she had encountered in her research a few days ago.

Bao Zi studied the mean of the Fourier coefficients of the self-defended L-function with a ternary quadratic variable.

According to the research plan established in the framework of the project.

To solve this problem, it is necessary to establish two functions with variables n, which represent the Fourier coefficients of the Maass pointed form and the fully pure pointed form, respectively.

Then, using Dirichlet's rational approximation theorem and Chauchy's inequality, T(-a; x) estimates over the principal interval and S1(a,√2) estimates over the remainder interval.

Bao Zi is stuck in this step.

To put it simply, Bao Zi did not figure out how to use the Fourier coefficient of Maass pointed form and pure pointed form to accurately obtain T(-a; x) estimates on the principal interval and S1(a,√2) estimates on the remainder interval.

"That's ......"

Gu Lu touched his chin and nodded understandingly.

Bao Zi is right, this place is indeed one of the difficulties of the subject.

Once it is not handled well, it is easy to lose all the efforts made.

However, this is not a problem for Gu Lu.

After speaking, Bao Zi took a bite of the bun, squinting comfortably, looking very satisfied.

"Well, after thinking about it all night, I don't have a clue at all, and it's very uncomfortable. ”

Bao Zi said vaguely, but there was no trace of annoyance on his face.

"Teacher, this problem, is it difficult for you, right?" Bao Zi stared at Gu Lu with bright eyes.

Gu Lu nodded.

Bao Zi smiled and said, "Then it's troublesome for the teacher to solve the puzzle." ”

Gu Lu smiled helplessly, and casually took a blank piece of scratch paper from the desktop.

He took out a carbon pen from the pen holder, and after a few seconds of silence, Gu Lu wrote six big characters on the paper.

"What's wrong with the ball?" Bao Zi sighed softly.

Gu Lu smiled faintly and said, "That's right, it's the whole problem in the ball." ”

The full name of the whole point problem in the ball is the prime distribution problem of the whole point in the ball.

This is one of the more well-known problems in the field of analytic number theory.

However, the issue has not yet been completely resolved.

However, although the whole point problem in the sphere has not been completely solved, it does not prevent mathematicians from using their knowledge to solve other mathematical problems.

For example, the problem at hand.

At present, Bao Zi encounters this problem, and it is not the only solution to solve it by using the whole point problem in the ball.

But after comparing several options, Gu Lu believes that this is the simplest solution.

And Bao Zi's side, after Gu Lu's reminder, suddenly realized.

There's a lot to know about the whole point problem inside the ball.

But there is only one knowledge related to the research content of the topic.

It was in the nineties of the last century, by two Chinese mathematicians using the ternary quadratic form, on the basis of the whole point problem in the sphere:

πΛ(x):=∑(m1^2+m2^2+m3^2≤x)Λ(m1^2+m2^2+m3^2)=8C3I3X^(3/2)+O(x^(3/2)log^(-A)x)

Of course, the prerequisite for this formula to be true is A>0.

The formula is not complicated, but it is one of several major research results on the whole point problem in the sphere.

Because it exposes the distribution of prime numbers at the whole point of the ball.

Although he vaguely guessed something, Bao Zi was not very sure, so he looked at Gu Lu with an inquiring gaze.

Gu Lu no longer sells Guanzi.

Write down a line of formulas on a piece of paper with a few clicks.

πΛ(x):=∑(m1^2+m2^2+m3^2≤x)Λ(m1^2+m2^2+m3^2)=8C3I3X^(3/2)+O(x^(3/2)log^(-A)x)

This formula is exactly what Bao Zi guessed.

However, Bao Zi did not speak rashly, but waited for Gu Lu's next words.

Gu Lu circled 'C3' and 'I3' in the formula and explained, "These two symbols, C3 represents the singular series in the whole point problem in the sphere, and I3 represents the singular integral, we can ...... like this first."

“…… On the basis of the above premise, the formula π3(x)=12C3I3∫t^0.5/logtdt+O(x^1.5log^(-A)x) can be obtained from the formula πΛ(x):=(omitted). ”

Gu Lu's speed of speaking was very fast, but Bao Zi next to him could easily keep up with Gu Lu's speed without the slightest pressure.

You can even take time to eat a few bites of steamed buns.

Gu Lu's train of thought, Bao Zi understood most of it.

To put it simply, it is to use the formula of the in-sphere integral problem of the ternary quadratic type to obtain the singular series and singular integral.

Then, on the basis of singular series and singular integrals, the formula for the mean problem related to the divisor function is obtained.

Sure enough, the last step that Gu Lu talked about was the derivation of the mean problem of the divisor problem.

“…… Finally, on the basis of the previous five formulas, we can derive a formula for the mean problem related to the divisor function, that is, ......"

Since there was no preparation in advance, Gu Lu calculated this formula on the spot.

After going through it briefly in his mind, Gu Lu wrote down the final formula on a piece of paper.

S(x):=∑(1≤m1，m2，m3≤x)d(m1^2+m2^2+m3^2)=8ζ(3)/5ζ(4)x^3logx+O(x^3).

"Hey, this formula ......"

When the full picture of the formula was presented in front of Gu Lu, it seemed that he had thought of something, and Gu Lu's pupils shrank suddenly.