I just want to be a quiet scholar

Chapter 263 Expressions

"Questions about number theory and functional analysis can be asked now. Mathematical questions in other fields, after the tutorial class is over, I welcome you to come to me at any time to discuss."

Shen Qi emphasized the rules, the 2-hour tutorial time should focus on number theory and functional analysis, otherwise, if you ask about geometry, he asks about topology, and the order of the class will be messed up.

"Okay, then I have no problem." Ajie shook his head. He is not very interested in number theory and functional analysis, but these two courses are compulsory courses for students in the mathematics department. All in all, it is good to get credits.

"Odd, I have a question about number theory." At this time, an Arab-faced student spoke. His name is Mohammed. According to him, his family was quite rich when he was a child, and he received a good education. Later, he became poor in the war , He was out of school for more than a year.

"Please express your doubts, Muhammad." Shen Qi stood in front of the blackboard and made a gesture of invitation.

Muhammad is a few months older than Shen Qi, he just joined Shen Qi's tutoring class this semester, unlike Ajie and other four people who worship Shen Qi so much.

Muhammad said: "I have consulted Riemann's manuscript, and he wrote in the manuscript: These properties of ζ(s) are derived from one of its expressions, but I have not been able to simplify this expression to be published in public form."

"That's right, this is Riemann's original text." Shen Qi nodded. He had studied Riemann's manuscript no less than a hundred times, and he knew it by heart.

"So strange, can you tell me, what kind of form should the unpublished expression mentioned by Riemann in the manuscript be?" Muhammad asked.

"Muhammad, this question is so wonderful!" Shen Qi's eyes lit up, and then he smiled helplessly: "If I can write this legendary expression, at this year's International Congress of Mathematicians, I can at least do A 45-minute report. Unfortunately, until today, I have not received an invitation letter from IMU."

"It seems that the proof work of Riemann's conjecture will continue for another 100 years." Muhammad spread his hands and said, he was slightly disappointed that he did not get a satisfactory answer.

Regarding the expression in Riemann's manuscript, Shen Qi really couldn't answer it. If he fiddled with this expression, he would not be far from proving Riemann's conjecture.

It is a mystery in the history of mathematics whether the expression "not simplified enough to be released to the public" that Riemann said really existed and in what form it existed.

The research on Riemann's manuscript has lasted for more than 100 years. Mathematicians believe that even if this expression is not simplified to the most perfect form, it is still of great significance for cracking RH. After all, this represents the core idea of ​​Riemann himself.

The god-level masters in the history of mathematics all have a small problem, that is, they like to play pranks. They perfectly pave the way for the foreplay that makes people want to burn, and then there is no aftermath.

The Fermat stem is "too little blank space in the book", and the Riemann stem is "not simplified to a form that can be published to the public."

Shen Qi has included these two stalks in his "History of Number Theory". Around Fermat's stalk, he will write a whole volume of "Fermat's Man and Fermat's Series of Conjectures".

The materials are already available, and most of the important and famous Fermat series conjectures have been proved. What remains is to spend time sorting out and write the history of number theory during Fermat's period as interestingly as possible without losing professionalism.

Shen Qi has also collected a lot of materials around the Riemann stalk, but the most critical issue is that the Riemann Hypothesis has not been fully proved, and the volume of "Riemann and the Riemann Hypothesis" cannot be finished perfectly, and may end with Unfinished way forced to end.

It is irresponsible to write the history of number theory without writing the Riemann conjecture and Goldbach conjecture.

However, all the books about Li Guai and Ge Guai in the world are just to introduce these two conjectures, let readers know the nature of Li Guai and Ge Guai, and there is no dry goods.

Shen Qi wanted to write something dry, but suffered from his limited level. In the end, his "History of Number Theory" may have become a mere formality, turning it into a book of hydrology.

After finishing the tutorial class, Shen Qi came to the Flint Library to look up Riemann's manuscripts again.

The Flint Library has 70 miles of shelves in the main library alone, all open.

Readers can enter the library to check the books and periodicals they need, from first-year freshmen to professors with Nobel Prize and Philippine Prize, from assistant staff to principals, all are treated equally.

Copies of ordinary books and published papers can be borrowed. Rare books, ancient books and manuscripts cannot be borrowed, but they can be read in the reading room.

The original version of the Riemann manuscript is the German edition, collected in Germany.

The PSU Flint Library houses the English version of the manuscript, which was translated by von Neumann himself during his tenure at PSU.

Von Neumann is an all-rounder. He understands mathematics, computers, nuclear weapons, and biological and chemical weapons, and he has reached the top level.

The English version of Riemann’s manuscript translated by von Neumann is only 8 pages long, and Shen Qi has already memorized it fluently. He will study it again today, carefully reading word by word, trying to figure out Riemann’s thoughts more than 100 years ago across time and space, and hope Can find even the slightest clue.

The Riemann conjecture has only one sentence: "All the complex zeros of ζ(s), that is, all the zeros of ξ(s) are on the straight line σ=1/2."

Riemann himself did not prove this conjecture. If he was to prove it successfully, it should now be called Riemann's theorem.

For the Riemann conjecture, Riemann gave an important prediction of the nature of ζ(s), the now well-known equation:

π^-s/2γ（s）ζ（s）=π^-（1-s）/2γ（1-s）ζ（1-s）

However, Riemann himself said: "These properties of ζ(s) are derived from an expression of it, but I have not been able to simplify this expression to a form that can be announced to the public."

So this expression that needs to be simplified is the key among the keys, and it is the key to cracking Riemann's conjecture.

"In fact, my previous thinking was not wrong. I tried to find the two recursive expressions of ζ(s), which coincided with Riemann's derivation logic. Riemann, Riemann, the unpublished one you mentioned What is the form of the expression?"

Shen Qi stayed in the library all night, and seemed to have gotten a little bit of inspiration.

But this inspiration is very ethereal, and Shen Qi can't write it down concretely. The deduction, imagination, and judgment of the 11th grade math level are still a bit low. It would be great if he can be promoted to the 12th level.

Back at the apartment, Shen Qi checked the submission system of "Mathematical Inventions".

"Recorded!"

Shen Qi was pleasantly surprised, recently his wishes have come true, he can do whatever he wants.

The paper "Shen's Proximity Theorem" has been officially included by "Mathematical Invention", one of the four major journals.

It will definitely be included. Professor Witten and Director Fefferman have both confirmed that there is no problem with "Shen's Proximity Theorem".

The two Philippine award winners said no problem, so it must be no problem.

System: "New achievement! Congratulations to the host for publishing a paper in "Mathematics Invention", one of the four major mathematics journals. The basic award is 5 Xueba points, multiplied by the journal's IF value of 3.331, and then multiplied by the mathematics master's talent The coefficient is 2.0, and the final reward amount is 333100 Xueba points. The balance is 2021400 Xueba points, please confirm with the host."

"These three 10,000+ Xueba points come just in time." Shen Qi expressed satisfaction.

At present, "Mathematical Invention" has included papers on "Shen's Proximity Theorem", but IMU has not yet officially recognized it.

After a while, IMU recognized the "Shen's Proximity Theorem", and there should be follow-up income from Xueba points.

"Mathematics is promoted to a higher level, and you will be promoted to a senior master." Shen Qi took out 1999862 points to upgrade mathematics to the 12th level.

To upgrade from level 11 to level 12 requires 200 Xueba points. Recently, Shen Qi gained 138 math experience points through daily experience accumulation, so 1999862 Xueba points are enough.

System: "Congratulations to the host for upgrading to level 12 in mathematics. The host's logical deduction ability, observation ability, judgment ability, imagination, memory and other indicators in the field of mathematics have significantly improved compared with the previous level."

Host, Shen Qi

age, 22

Math Level 12, 0/400

Physics Level 6, 1123/15

Sports Level 5, 3019/5

English Level 5, 987/5

Chinese Level 5, 999/5

Political Level 5, 1360/5

Chemistry Level 3, 502/5000
Biology Level 3, 520/5000
History Level 2, 1139/3000
Geography Level 2, 689/3000
The balance of Xueba points: 21538 points

Level 11 is a master, level 12 is an advanced master, level 13 is a top master, level 14 is almost invincible, and level 15 is so lonely.

Shen Qi, who was successfully promoted to a senior master of mathematics, suddenly got that illusory inspiration, which seemed to be concrete: "I thought of it! I thought of it! Riemann in 1859, he probably wanted to deduce it like this..."

(End of this chapter)