In the office, Xu Chuan clicked on the link sent by Xiao Ling with great interest and downloaded the paper in the email.

When he saw the title of the paper, he raised his eyebrows with a look of interest in his eyes.

"New Large Value Estimation Based on Dirichlet Polynomials"

The title of the paper is very simple, but when it comes to the study of the Riemann Hypothesis, nothing is that simple.

Dirichlet-polynomial distribution is a probability distribution, which is a generalization of polynomial distribution. This mathematical tool is generally widely used in probability theory and statistics, and is often used in natural language processing, text mining and other fields.

For example, the latent Dirichlet allocation (lda) algorithm is used in topic models. It also plays an important role in Bayesian statistics and is used to describe multi-category random variables.

In addition, it is also used to describe the probability distribution of multiple mutually exclusive, discrete results in an experiment.

For the Riemann Hypothesis, the Dirichlet polynomial bound plays an important role in several issues related to the distribution of prime numbers.

Simply put, they can be used to limit the number of zeros of the Riemannian zeta function in the vertical strip, which is related to the distribution of prime numbers in short intervals.

That is: the Dirichlet polynomial can be expressed as: "D(t)=\sum_{n = N}^{2N} b_n n^{it}."

But to be honest, using this tool to study the Riemann Hypothesis is not a very new thing.

Decades earlier, in 1940, mathematician Professor Albert Ingham had used this tool to establish classical bounds on the zeros of the Riemann zeta function and, more generally, on the large values ​​governing various Dirichlet series.

Substantial improvements have been made.

However, in the following decades, the inferences about the Riemann Hypothesis were limited to this, and there has been no breakthrough.

Therefore, Xu Chuan is still looking forward to the paper in his hand.

This may provide him with some value in researching the Riemann Hypothesis.

After all, if it had no value, the editor-in-chief of "New Advances in Mathematics" would not be able to personally send the paper to him and invite him to review it.

In the office, Xu Chuan quickly downloaded the paper and sent it to the printer, ready to print it out.

Compared with reading papers directly on the computer screen, he prefers to use paper manuscripts.

Before that, he clicked on the paper on the computer and started browsing it eagerly.

"...It's a bit interesting. The first part of this paper is actually based on Fourier analysis, but it does not use the traditional stationary phase method."

"It is based on the new frequency limit of the large value of the Dirichlet polynomial, which substantially improves the zero point limit of the Riemann zeta function given by Ingham..."

Reading the paper in his hand, Xu Chuan's eyes were full of interest and thinking.

I have to say, this is indeed a very clever method.

For using Dirichlet polynomials to express the Riemann function, the most important part is the size of the D (t) hyperlevel set.

The author of this paper normalizes so that the coefficient norm is at most 1, and then studies the hyperlevel set | D (t) |> N^\sigma, where the sigma index is between 1/2 and 1.

This alone is enough to reflect the brilliance of this paper.

While flipping through the papers, Xu Chuan muttered softly.

A paper with one exciting point is enough for people like him.

"Professor, your printed paper is ready."

While Xu Chuan was immersed in his papers, the door of the office was lightly knocked twice, and assistant Lu Ling walked in quickly with a stack of papers in her hands.

"give it to me."

Xu Chuan reached out to take the paper without hesitation and ignored Lu Ling.

Just when he was about to learn more about this paper, he suddenly remembered another thing and shouted.

"Xiao Ling, help me reply an email to Professor Robert Morey Dean, the editor-in-chief of "New Advances in Mathematics", saying that I have accepted the review invitation."

In the lower right corner of the display screen, a chat box that was originally hidden popped up.

"Copy that, Master!"

“Email replied!ヾ(≧▽≦*)o”

At the same time, at the door of the office, Lu Ling, who was about to turn around and go out, was stunned when she heard the sound and stopped.

She looked at Xu Chuan with some surprise and confusion.

Is this... giving her an order?

But her name is not Xiao Ling, and the professor seems to have never called her that.

After hesitating for a while, Lu Ling decided to ask. After all, what if the professor got angry and shouted more intimately?

"Um...Professor, did you just ask me to send an email to the editor-in-chief of "New Advances in Mathematics"?"

Hearing the voice, Xu Chuan replied without raising his head: "I didn't call you. I was talking to someone else. It's okay."

Lu Ling was a little confused, but she still replied: "Okay, if you need anything, just tell me."

While replying, she also looked around the entire office.

In broad daylight.

Who is the professor talking to?

If you didn’t call her, who would you call her?

Could this be...haunted?

Thinking of this terrible possibility, Lu Ling couldn't help but shudder.

Although as an assistant to a great scientist, she shouldn't believe in such things as metaphysics.

But she has been afraid of ghosts since she was a child.

It is broad daylight, there is no one else in the office, but the professor is talking to others. This... this is too scary...

After muttering something, Lu Ling walked out quickly.

"Tang Ran, Tang Ran. Let me tell you, there might be a ghost in the professor's office?"

Lu Ling approached another assistant, who was now her best friend, and spoke in a low voice.

In the assistant's office, Tang Ran, who was sorting out information, was stunned and looked at his good sisters suspiciously.

"What's wrong with you? Are you possessed?"

While she was talking, she reached out and tested Lu Ling's forehead to see if she had a fever, which was so high that she started talking nonsense.

Lu Ling slapped her hand away and whispered quickly: "I'm serious, the professor was talking to people in the office just now, and even gave instructions to work. But there was no one else in his office."

"What if this isn't haunted?"

Hearing this, Tang Ran didn't know whether to laugh or cry. He pushed her head angrily with his finger and said, "What's in your head?"

"This is the professor's office, how could there be such a thing as being haunted?"

"But...but there is really no one else in the professor's office." Lu Ling quickly added.

Tang Ran said helplessly: "Maybe the professor is chatting with someone else? You are an assistant to a great scientist. You believe in ghosts. You are not afraid of people laughing at you if you tell me."

Lu Ling muttered, and Tang Ran had no choice but to say, "Okay, it really can't be done. Just ask the professor after he gets off work."

.......

In the office, Xu Chuan still didn't know that the tasks he casually assigned to the AI ​​assistant were misunderstood by his assistant as talking to a ghost.

At this moment, he was already immersed in the paper in his hand.

Although "New Advances in Mathematics" was sent to him, the overall paper that he was invited to serve as a reviewer was not long, only a dozen pages.

But I have to say that this paper has indeed opened up a new direction in the study of the Riemann Hypothesis.

Normalize the size of the D (t) hyperlevel set so that the coefficient norm is at most 1, then study the hyperlevel set | D (t)|> N^\sigma, and improve the sigma limit from 1/2 to

Close to 3/4.

In addition, the paper also discusses existing simple estimation methods and their limitations by analyzing the new bounds on the large values ​​of Dirichlet polynomials and the magnitude of the Dirichlet polynomial norm on a specific set.

These studies are of great significance in analytic number theory. They have created new tools for studying the size of the Dirichlet function on specific sets, which can solve similar problems more cleverly and simply.

What Xu Chuan regrets is that this paper did not advance the Riemann Hypothesis much.

In other words, this paper just advances the distribution of non-trivial zero points of the Riemann function through another route based on the original basis.

But there is still a long way to go before he can prove the weak Riemann hypothesis, and it has not broken through the limit he set.

This is also what Xu Chuan regrets.

...

He carefully read the paper from beginning to end, and reviewed it carefully. After confirming that there were no problems, Xu Chuan turned on the computer, wrote down his review comments at the end of the paper, and marked 'passed review'

'Four words.

Generally speaking, the review of papers is a relatively tedious task.

Even for ordinary papers, in most cases the review will take three to five days or even longer to complete.

After all, for academia, rigor is the most important spirit. Ensuring the correctness of reviewed papers is the basic principle of every professor invited to review papers.

For papers related to the Millennium Problem such as the Riemann Hypothesis, the review time will only be longer.
To be continued...