New Approaches to the Riemann Hypothesis
The seven millennium problems in mathematics are truly difficult to solve.
Currently, only the Poincaré conjecture has been solved. Russian mathematician Perelman, Lü Qiujian, completely proved the Poincaré conjecture.
The Riemann Hypothesis was proposed in the 19th century, spanned the entire 20th century, and remains unchallenged in the 21st century.
Any mathematician studying number theory would be eager to prove RH, which would be a monumental achievement that would go down in history.
Just as difficult as the proof of Goldbach Conjecture, RH has not been fully proven even after three centuries.
Goldbach Conjecture of 1+1 has not been proven, but Mr. Chen Jingrun proved 1+2, which is the closest result to Goldbach Conjecture.
Proving in one step that RH and Goldbach Conjecture are not easy things to do is a testament to history.
Mathematicians have been working on interim proofs of RH for centuries.
Regarding the formula for the Riemann zeta function ζ(s), for any complex number, if Re(s) > 1, then:
ζ(s)=Σn^-s=∏(1-p^-s)^-1
Where n is a natural number and p is a prime number.
Mathematicians have tried every possible method and means, from Euler's classical formula to Bernoulli numbers, and then to Ramanujan Formula for positive odd numbers, and have finally made important progress. The special cases of k=3,5 and k=4,6,7 have been recognized by all contemporary mathematicians.
Currently, there is still one bridge between the progress made and the complete proof of RH.
The crucial bridge connecting the preceding and following steps is the two recurrence formulas of ζ(2n+1).
If the two recurrence formulas of ζ(2n+1) can be proven, then Shen Qi believes that Professor Muller's team is not far from finally proving RH.
What Shen Qi was that the unfinished paper in his hand was a discussion and proof of the two recurrence formulas of ζ(2n+1).
The framework of this paper was set by Muller, while the specific arguments and proofs were written by Mary.
It is obvious that Professor Muller's strategic direction is correct, but Mary's tactical execution has had little effect.
Mary's tactics are too outdated. With your logic of calculation and derivation, RH should have been completely proven long ago, but that's not the case. Shen Qi returned the paper to Muller, saying, "We need a new lemma to prove that the conclusion holds when k=1. Then, the two recurrence formulas for ζ(2n+1) can be reasonably derived, thus launching a full-scale attack on the complete proof of RH."
"Hey kid, I've thought about that too!" Muller's eyes lit up as he looked at Shen Qi.
"Us?" Mary looked at Shen Qi questioningly, then said matter-of-factly, "Yes, us. This is a research project that Allen and I are working on together."
"Mary, I have a bold idea. We could invite Shen Qi to join our team to study the topic of ζ(2n+1). What do you think?" Muller was very democratic; he politely asked his student Mary.
"I think we should maintain the status quo, because there's nothing wrong with it," Mary said with a strange expression.
"I would be happy to Professor Muller's ζ(2n+1) research group." Shen Qi ignored Mary's questioning expression and directly expressed his determination to Muller.
"Your main focus is Mathematical Physics, with a secondary focus on Algebraic Geometry. Shen, don't tell me you want to add number theory as well," Mary said coldly.
"Professor Muller is my idol. He is proficient in multiple fields such as Mathematical Physics, Algebraic Geometry, number theory, and Group Theory. My second minor choice of number theory will not be a bad thing for me or the whole team," Shen Qi replied.
"Shen Qi, if your girlfriend agrees to you minoring in number theory, I have no problem with it. I've told you before why you didn't choose number theory, after all, you're a student Elon praised." Muller intended for Shen Qi to collaborate with Mary.
"She will agree." Shen Qi couldn't miss the opportunity to work with the Princeton University Mathematical Research team to tackle RH, Ou Ye would definitely agree and support him in doing so.
"Mary, why not give Shen Qi a chance? You need his help. You yourself admit that you cited Shen Qi's algorithm in your doctoral dissertation; you've actually collaborated once before," Muller said with a smile. "Shen Qi is an outstanding student; he published ten mathematical papers in two years. Mary, the number of papers you and I published during our undergraduate years combined is exactly half of Shen Qi's."
Mary remained expressionless but ultimately agreed to Muller's proposal, after all, Muller was the Boss of the team.
"So, Shen Qi, from now on you are Mary's assistant. I wish you a pleasant working relationship." Muller gave the final instruction and adjourned the meeting.
Shen Qi obtained some information on the ζ(2n+1) problem. He was ambitious and had a well-developed idea: he would use an innovative approach to prove the two recurrence formulas of ζ(2n+1).
"You'd better not cause me any trouble." Mary invited Shen Qi to have coffee at the school's cafe so they could be alone and discuss the next steps for their research project.
"Mary, do you think I'm the kind of person who would do something reckless?" Shen Qi retorted.
"Who knows?" Mary remained noncommittal.
"If I'm someone who acts recklessly, then so are you. Your doctoral dissertation cited my algebraic approximation algorithm. If you had personally developed that algebraic approximation algorithm, you might have just graduated with your doctorate now, or you might not graduate until next spring. That algebraic approximation algorithm can't be calculated in less than a few months." Shen Qi stated the facts; the Diophantine equation Wash Conjecture paper incident had always been a sore point for him.
"The paper on the Diophantine equation Wash Conjecture is already in the past," Mary said casually. "So, tell me, how do you plan to proceed with the work on the ζ(2n+1) topic?"
"Professor Muller's framework remains unchanged, but the specific arguments prove to overturn it and start over," Shen Qi said firmly.
"Are you sure?" Mary sneered.
"Absolutely certain." Shen Qi nodded resolutely. "Mary, I know you're very busy, teaching undergraduate Recitation Class and grading their assignments and exams. If you don't mind, I will complete the new research on the ζ(2n+1) topic on my own. I will finish this work before Christmas. Don't worry, your name will be listed as a co-first author."
"Hahaha." Mary laughed so hard her stomach hurt, as if she had heard the funniest joke in the world.
"Arrogant, ignorant, and beyond redemption." Mary took a ten-dollar bill from her elegant purse, placed it under her coffee cup, and got up to leave.
"Coffee is on me," Mary said, turning back after taking a few steps. "But there are only two co-first authors, Professor Muller and I."
"Then let's each use our own methods," Shen Qi said, spreading his hands.
"Hmph." Mary walked away coldly; her cooperation with Shen Qi did not seem pleasant.
Shen Qi published a total of ten papers during his time at Yenching University. It was this rich research experience that made him the only graduate student in China to enroll in the Mathematics Department at Princeton University this fall.
Shen Qi is either the first author or a co-first author of all ten of these papers.
A paper can have more than one author. The authors listed below the paper title are the first author, counting from left to right.
Shen Qi had marked his own name, as well as those of Ou Ye, Sun Erxiong, and Lu Guozhen, with the marks indicating their status as co-first authors.
Then come the second author, the third author...
If you're not aiming to be the first author or co-first author when writing a paper, what's the point of writing it at all?
Shen Qi could not accept that his name was listed as the first author, or after the co-first author.
Mary disliked Shen Qi, so Shen Qi had no choice but to go it alone.
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