The top student must be diligent.

Chapter 116: Riemann hypothesis is not worth mentioning?
Seeing the name, Faltings silently moved the cursor on the screen to the PDF link next to the title and clicked on it.

I entered the PDF interface of the paper, downloaded it, and then opened the PDF file viewer to read it.

"It turned out to be Xiao Yi's paper... Isn't he in UCLA now? How come he still has the energy to publish a paper?"

Faltings couldn't help but have such thoughts in his mind.

And the result is such an important one.

"Abstract...According to the classification sieve method, the self-conservation theory of Etale algebraic varieties is inverted to extract the basic group information from the L-function...After analytical extension, a polynomial expansion in the form of Fourier expansion is performed...This paper mainly demonstrates its use in the approximation of the critical line of the Riemann hypothesis, and finally successfully approximates the critical line theorem to 50%, but I believe there is still room for improvement."

The critical line theorem of the Riemann hypothesis, which had not been broken through for a long time, was directly raised to 50%!

Faltings dares to guarantee that this result can be said to have a stimulating effect on the entire mathematical community, even though it does not actually prove the Riemann hypothesis.

Because it means that after more than thirty years, the critical line approximation method is still a method that can be used to try to solve the Riemann hypothesis.

It can re-inspire the mathematical community to further explore the critical line approximation method and encourage more people to try it.

This is the Riemann hypothesis!
Even the slightest possibility would arouse the enthusiasm of the entire mathematics community.

As for last year's paper, it is not important, because the improvement from 40% to 41.7% is indeed not a huge achievement. Faltings has also read that paper, and the method used in it is not that impressive. In fact... he can know what the method is just by reading the abstract.

He had come up with this result out of boredom, but he didn't like it, so he threw it aside a long time ago.

Only the breakthrough results in Xiao Yi's paper can truly illustrate that as long as new mathematical tools are continuously developed, they can help them achieve further results in the approaching line approximation method.

However, when Faltings finished reading the summary, he didn't know why he saw a kind of...

The feeling of Versailles.

Such a breakthrough in the Riemann hypothesis, but Xiao Yi only used it as a case demonstration?
Think his polynomial expansion is more powerful?
If this is not Versailles, then Faltings doesn't know what Versailles is.

"I want to see how important your new polynomial expansion is. It even outshines the Riemann hypothesis."

Faltings couldn't help but become curious, so he turned the page and began to read the main text.

This paper is not long, only 21 pages. Excluding the space occupied by citations, there are only 20 pages of main text.

At the same time, only the first 8 pages are used to introduce the new polynomial expansion, and the remaining 12 pages are the approximation of the critical line of the Riemann hypothesis. Of course, this aspect does not require too much content. Kang Rui's paper at that time was only 26 pages.

However, Faltings was very clear that the content of the first eight pages was the core of the paper.

"Yeah, it's very short. You can finish it in about half an hour."

This is what Faltings thought at first, but as he started reading, he realized that these 8 pages might contain more technical content and be more complex than the 20 pages of other papers.

He also read it much more carefully and cautiously than any other paper, for fear of missing any small point.

Until two hours later.

After reading 8 pages, Faltings finally finished reading.

The desk was already piled with draft papers, all of which were used to verify the contents of these 8 pages, and his face was full of disbelief.

"This polynomial expansion... is actually..."

How to describe this new polynomial expansion?
Faltings can only say that in the field of complex analysis, its role is no less than Taylor expansion, Fourier expansion, etc.

Regardless of whether it is Taylor expansion or Fourier expansion, etc., their status in the mathematical world is unquestionable. For example, Taylor expansion, as the most basic method in calculus, exists almost in the entire mathematical world. No mathematician will say that he does not understand Taylor expansion, which will be ridiculed.

And now, Faltings has seen this new development and its potential.

At least over the domain of complex numbers, this is obvious.

"It can more fully display the information hidden in some algebraic expressions in the complex number domain, which can be used to help analyze almost all problems in mathematics!"

Finally, Faltings understood why the Riemann hypothesis could only be used as a case study in this paper.

Indeed, compared with this new polynomial expansion method, the Riemann hypothesis is nothing at all. It is just the 50% critical line. In the face of such a heavyweight result, it is not worth mentioning at all.

Unless it's Riemann's theorem.

Faltings couldn't help but go back to page 1, intending to read the eight pages again from beginning to end.

As for what was on the next 12 pages, he no longer cared about it, because after reading these 8 pages, he already knew what was written on the next 12 pages.

In this way, Faltings spent nearly an hour reading the eight pages again.

Finally, he leaned back tactically and leaned on the back of his office chair.

"That kid... has come up with something so incredible again, really..."

“Is there any limit to his mathematical ability?”

He smiled helplessly, and then began to think.

Using this new development...well, let's name it the Xiao development.

Using Xiao's words, it seems that it can also bring some help to his recent research.

As a top mathematician, the problems he studies are basically the most cutting-edge problems in mathematics, including the Riemann hypothesis.

Of course, he and Xiao Yi held the same idea that critical line approximation was not the right way, so he had been thinking about the possibility of using other methods to solve the Riemann hypothesis.

Xiao's development seemed to help him think about the possibility in this regard. Oh, and there is also the Landau-Siegel zero-point conjecture.

Faltings remembered that Zhang Yitang was studying this problem.

From this point of view, Zhang Yitang is lucky. With Xiao Yi's new method, solving the Landau-Siegel zero-point conjecture seems to be much easier, especially since they are going to cooperate...

Faltings estimated that it would not be long before the Landau-Siegel zero-point conjecture was solved.

“Is this the benefit of having a talented collaborator?”

Suddenly, there was a knock on the office door.

Schultz walked in.

He walked up to Faltings with two cups of coffee in his hands, put one of the cups on his desk, took a sip, and asked, "I was wondering why I didn't see you today. It turns out you stayed in the office all morning. What were you doing?"

“Read a paper.”

"Oh?" Schultz raised his eyebrows, glanced at the computer screen, and said, "That must be very long."

"It's not long, only 21... well, maybe 8 pages."

"8 pages? Such a short number, and you can actually spend a whole morning reading it?" Schultz was a little surprised.

Faltings stared at Schultz and took another sip of coffee before saying, "Xiao Yi wrote this."

"Walter cough cough cough..."

Schultz's eyes widened when he heard the name, and then he choked on the coffee he had just drunk.

Seeing this scene, Falting laughed so hard that he fell backwards. The prank was successful!
Let this kid use what happened in Princeton to threaten me?
He should know what it means to respect the elderly and love the young!
Although he was teased, Schultz didn't have time to dwell on it. When his throat felt better, he immediately asked, "When did Xiao Yi write the paper?"

"Hmm...let me take a look."

As Faltings spoke, he glanced at the publication time of the paper and then replied: "It was at nine o'clock this morning. I just opened arxiv today and it came out."

"You didn't even tell me when you saw it!"

Schultz complained, then immediately moved closer and looked at the paper.

The title of the paper also immediately surprised him.

"He has actually started studying the Riemann hypothesis now?"

"Besides... it's only been a short time since the Princeton conference! Isn't he with Terence Tao now? How did he push the critical limit theorem of the Riemann hypothesis so far?! Aren't they studying the x^2+1 prime number problem?"

After he finished reading the summary, he narrowed his eyes.

"Wait, this abstract... why does it give people the feeling that he doesn't take this breakthrough in the Riemann hypothesis very seriously?"

"Is his polynomial expansion the most important?"

"You figured it out too." Faltings stood up with a smile, picked up his cup of coffee, drank it in one gulp, and then said, "Take your time to read it. I'm going to have lunch first."

Faltings left, leaving Schultz alone in the office.

He sat in front of the computer and began to read the paper carefully, just like Faltings just now.

When Faltings came back, he saw Schultz scratching his enviable long hair, muttering: "This unfolding... this unfolding... My God, how did he come up with it?"

"How's it going?"

Faltings laughed.

Schultz came back to his senses, turned his head and looked at Falkins, and immediately pointed at the paper on the computer screen and exclaimed: "This polynomial expansion is really incredible! Although I have studied his Etale algebraic variety automorphism theory many times, as well as his classification sieve, I never thought that it could be used in this way!"

"He actually did such in-depth research on complex analysis!"

"Yes." Faltings nodded slightly.

"He always brings great surprises to the mathematical world, doesn't he?"

"I think that if the Riemann hypothesis is really proven in the future, this new polynomial expansion will definitely play an extremely important role in it."

Schultz did not refute Faltings' comments.

There is no need to refute.

Because he saw it too.

Perhaps this method will become the most immortal method in the entire field of complex numbers, just like the analytic extension in complex analysis!
……

After Faltings and Schultz finished reading this short 21-page paper, more and more people in the mathematics community had discovered this paper that had quietly appeared.

Just like the two of them, these scholars in the mathematics community were initially very curious about the title and abstract of Xiao Yi's paper.

The expansion of this polynomial must be so awesome that the major breakthrough in the Riemann hypothesis can become just a bonus.

Similarly, after they finished reading the first eight pages, all that remained in their hearts was a deep sense of disbelief.

Can mathematics be studied in this way?
(End of this chapter)