Expression
"You can ask questions about number theory and Functional Analysis now. As for other mathematical problems, you are welcome to come and discuss them with me at any time after the Recitation Class."
Shen Qi emphasized the rules: the two-hour Recitation Class should focus on number theory and Functional Analysis; otherwise, if everyone keeps talking at once, asking about geometry and topology, the class will fall apart.
"Okay, then I have no questions." Ajay shook his head. He wasn't very interested in number theory and Functional Analysis, but these two courses were required for the Mathematics Department students. In short, he just needed to get the credits.
"Qi, I have a question, about number theory." At this moment, an Arabia student spoke up. His name was Muhammad. According to him, his family was quite wealthy when he was young, and he received a good education. However, war broke out and they became poor, so he had to drop out of school for more than a year.
"Please state your question, Muhammad." Shen Qi stood in front of the blackboard and gestured for him to speak.
Muhammad was a few months older than Shen Qi. He had just joined Shen Qi's tutoring class this semester, unlike A-Jie and the other three who idolized Shen Qi.
Muhammad said, "I have consulted Riemann's manuscripts, in which he wrote: 'These properties of ζ(s) are derived from one of its expressions, but I have not been able to simplify this expression to a form that can be published.'"
"That's right, this is Riemann's original text." Shen Qi nodded. He had studied Riemann's manuscripts no less than a hundred times and knew them by heart.
"That's strange. Can you tell me what form that unpublished expression that Riemann mentioned in the manuscript should take?" Muhammad asked.
"Muhammad, that's a brilliant question!" Shen Qi's eyes lit up, then he smiled helplessly, "If I could write out this legendary expression, I could give at least a 45-minute presentation at this year's International Congress of Mathematicians. But unfortunately, to this day, I haven't received an invitation from IMU."
"It seems the work to prove the Riemann Hypothesis will take another 100 years," Muhammad said, shrugging slightly, somewhat disappointed that he hadn't received a satisfactory answer.
Shen Qi couldn't answer the question about the expression in Riemann's manuscript. If he could figure it out, he would be close to proving the Riemann Hypothesis.
Whether the expression Riemann described as "not simplified enough to be publicly available" ever truly existed, and in what form it existed, remains a mystery in the History of Mathematics.
The study of Riemann's manuscripts has continued for over a century. Mathematicians believe that even if the expression is not simplified to its most perfect form, it is still of great significance for deciphering the Riemann Hypothesis, since it represents the core ideas of Riemann himself.
The greatest mathematicians in History of Mathematics all had one quirk: they loved to play pranks. They would perfectly set the stage for something arousing, and then... nothing.
The Fermat trope is "too little white space in the book," while the Riemann trope is "not simplified to a form suitable for public display."
Shen Qi has already included these two memes in his book "History of Number Theory". He will write a whole volume on "Fermat: The Man and Fermat's Conjectures" around the Fermat meme.
The material is already available, and most of the important and famous Fermat conjectures have been proven. What remains is to spend time organizing and writing an interesting yet professional account of the history of History of Number Theory during Fermat's time.
Shen Qi has also collected a lot of material about the Riemann Hypothesis. However, the most crucial problem is that the Riemann Hypothesis has not been fully proven. The volume "Riemann: The Man and the Riemann Hypothesis" cannot be concluded perfectly and may end in a forced and unsatisfactory way.
Writing History of Number Theory without mentioning the Riemann Hypothesis and Goldbach Conjecture is irresponsible.
However, all the books in the world about the Lê Conjecture and Goldbach Conjecture simply introduce these two conjectures, letting readers know Lê Conjecture Goldbach Conjecture, without providing any substantial information.
Shen Qi write something substantial, but due to his limited abilities, his "History of Number Theory" may have become merely a formality and turned into a low-quality, superficial work.
After finishing his Recitation Class, Shen Qi went to the Flint Library to review Riemann's manuscripts once again.
The bookshelves in the main building of the Flint Library alone stretch for 70 miles, and are all open shelves.
Readers can access the library to find the books and periodicals they need, and everyone is treated equally, from first-year students to Nobel and Fields Medal professors, from support staff to the principal.
Regular books and photocopies of published papers can be borrowed, but rare books, ancient books, and manuscripts cannot be borrowed, although they can be read in the reading room.
The original Riemann manuscript was in German and is held in Germany.
Princeton University's Flint Library holds the English version of the manuscript, which was translated by John von Neumann himself during his time teaching at Princeton University.
John von Neumann was a polymath; he was knowledgeable in mathematics, Computer Science, nuclear weapons, and biological weapons, and he excelled in all of them.
The English translation of Riemann's manuscript by von Neumann is only eight pages long, Shen Qi can already recite from memory. Today, he is studying it again, carefully reading each word, trying to understand Riemann's thoughts more than a hundred years ago, hoping to find even the slightest clue.
The Riemann Hypothesis can be summarized in one sentence. "All the complex zeros of ζ(s), that is, all the zeros of ξ(s), lie on the line σ = 1/2."
Riemann himself did not prove this conjecture; if he had successfully proven it, it would now be called the Riemann Theorem.
Regarding the Riemann Hypothesis, Riemann gave an important prediction of the ζ(s) property, namely the now-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 one of its expressions, but I have not been able to simplify this expression to a form that can be published."
Therefore, this simplified expression is the key to solving the Riemann Hypothesis.
"Actually, my previous line of thought was not wrong. I tried to obtain two recursive expressions for ζ(s), which coincided with Riemann's derivation logic. Riemann, what exactly is the form of this unpublished expression you mentioned?"
Shen Qi spent the entire night in the library, seemingly gaining a little inspiration.
However, this inspiration was very elusive, and Shen Qi was unable to put it into concrete form. His reasoning, imagination, and judgment at level 11 in mathematics were still a bit low. It would be great if he could reach level 12.
Back in his apartment, Shen Qi checked the submission system for <Inventiones Mathematicae>.
"Already included!"
Shen Qi was pleasantly surprised; lately, everything he wished for had come true.
The paper "Shen Proximity Theorem" has been officially included in "Inventiones Mathematicae," one of the Top Four Journals.
It will definitely be included. Professor Witten and Director Feierman have both confirmed that there is no problem with "Shen Proximity Theorem".
If the two Fields Medal winners say there's no problem, then there definitely isn't a problem.
System: "New achievement! Congratulations, host, for publishing a paper in <Inventiones Mathematicae>, one of the four major mathematics journals. The basic reward is 50,000 Academic Excellence Points. Multiplied by the journal's Impact Factor (IF) of 3.331, and then by the Mathematical Talent Coefficient of 2.0, the final reward is 333,100 Academic Excellence Points. Remaining balance: 2,021,400 Academic Excellence Points. Please confirm."
"These 300,000+ academic points came at just the right time," Shen Qi said with satisfaction.
The paper on "Shen Proximity Theorem" has been included in the journal <Inventiones Mathematicae>, but it has not yet been officially recognized by the IMU.
After IMU acknowledges the "Shen Proximity Theorem" in a while, there should be subsequent academic credits earned by top students.
"Mathematics level up, promoted to Advanced Master." Shen Qi used 1,999,862 academic points to upgrade his mathematics to level 12.
Upgrading from level 11 to level 12 requires 2 million academic points. Shen Qi has recently accumulated 138 math experience points through daily experience points, so spending 1,999,862 academic points will be enough.
System: "Congratulations, host, your math level has been upgraded to level 12. Your logical reasoning, observation, judgment, imagination, and memory in the field of mathematics have all significantly improved compared to the previous level."
Host, Shen Qi
Age: 22
Mathematics Level 12, 0/4 million
Physics Level 6, 1123/150,000
Sports Level 5, 3019/50,000
English Level 5, 987/50,000
Chinese Language Level 5, 999/50,000
Politics Level 5, 1360/50,000
Chemistry Level 3, 502/5000
Biology Level 3, 520/5000
History Level 2, 1139/3000
Geography Level 2, 689/3000
Remaining Academic Points: 21538
Level 11 is Master level, level 12 is Advanced Master, level 13 is Top Master, level 14 is almost invincible, and at level 15, even with my invincibility, I feel so lonely.
Shen Qi, who had successfully advanced to the rank of Senior Mathematician, suddenly realized that his elusive inspiration could now be concretized. "I've got it! I've got it! Riemann in 1859, he probably wanted to derive it like this..."