I Want to See the Equations

"In 1859, Riemann published a paper entitled 'On the Number of Primes Not Greater Than a Given Value,' which was only eight pages long. This was his only publicly published paper on number theory."

"It was these mere eight pages that laid the foundation for Analytic Number Theory."

"It is clear that to be remembered in history, one does not necessarily need a large number of words; the quality of the article always comes first."

"We don't know what reason Riemann had for making such a conjecture in 1859; perhaps it was a stroke of genius."

"RH is equivalent to saying that all the zeros of Ξ(ω) are real."

"Riemann added that, of course, this needs to be proven, and he had done so, but because a core expression was not simplified to a publicly available level, it was not published. This is one of the greatest mysteries in the History of Number Theory."

"You can find similar statements in any mathematics book or paper, but what I am about to describe is original content published for the first time..."

Shen Qi is writing his "History of Number Theory" with great passion. With substantial content, his enthusiasm for writing is truly high!

Let the set of non-obvious zeros of the Riemann Zeta function be:

{ρ1,1-ρ1,ρ2,1-ρ2,...,ρk,1-ρk,...ρn,1-ρn}

The set is illustrated as follows:

Any two non-obvious zeros that have the characteristic of 'summing up to 1 and having the same absolute value of their imaginary parts' are matched as a pair.

For ease of reference, the author will refer to this new processing method as the 'twin matching method'.

Below, I will derive the core expression of ζ(s) using the 'twin matching method'.

Shen Qi wrote furiously. He was about to derive the core expression of ζ(s) and prove the Riemann Hypothesis. If he could do that, his book, "A History of Number Theory," would definitely be a bestseller, and he would become a legend with just one book!

The "twin matching method" is an inspiration that Shen Qi just came up with; it's his original creation.

The numbers game will eventually come to an end, and Shen Qi decided to end the Riemann Hypothesis game.

Too excited to sleep, Shen Qi worked until dawn.

"Therefore, under the 'twin matching method', the core expression of ζ(s) should be: ζ(s) = e^A + Bs∏∞n=1(1-s/ρn)(1-s/1-ρn)e^(s/ρn + s/1-ρn)... So that's how it is..."

Shen Qi stood up, stretched his limbs, and looked calmly at the rising sun outside the window, then smiled.

The numbers game isn't over, but Shen Qi has found the right path, which is a very important breakthrough.

"Therefore, the unpublished expression mentioned by Riemann is not one, but two, or even three. The word 'one' is inappropriate; it should be 'a set.' To fully prove the Riemann Hypothesis, a set of core expressions is needed."

Shen Qi worked through the night and discovered a huge secret: the whole world had been fooled by Riemann for over a hundred years.

Whether Riemann's mistake was a slip of the pen or a deliberate error is something no one can say for sure.

Shen Qi immediately told his girlfriend this huge secret over the phone, "I think I've found a solution to RH. I've created a new method, and I'll tell you about the basic setup..."

"Words are no proof, I want to see the formula!" Ou Ye exclaimed excitedly on the other end of the phone.

Shen Qi: "If we follow my 'twin matching method' setting, the core expression for proving RH should have one set, and I've already derived one of them. Next week I plan to visit Professor Gong at Columbia University; he's a top expert in this field, and I'd like to hear his opinions and suggestions. I'll show you the formula next week; I need to refine it."

Ou Ye: "You said you'd come see me in a month, and it really is a month. Okay, see you next week."

After hanging up the phone, Shen Qi slept for a while. After getting up, he continued to derive another theoretically existing expression.

However, the problem is that, based on the "twin matching method" and the first expression, the second expression cannot be obtained no matter what.

"Sigh, I've tasted a little bit of the sweetness, but I want it all. I wish I could reach level 13." Shen Qi looked at the system. Leveling up from 12 to 13 requires 4 million academic points, which means publishing more than 10 papers in the Top Four Journals.

From September of the year before last until now, a year and a half has passed, and Shen Qi has only published three papers in the Top Four Journals. It will take several more years for him to publish ten new papers in the Top Four Journals.

"I need to listen to the opinions of the masses, make full use of Princeton's superior resources, and win this important battle through teamwork."

Shen Qi walked toward Professor Muller's office, knowing that Professor Muller was of sound mind.

A year ago, Muller arranged for Shen Qi and Mary to work together on a project to derive two recursive expressions for ζ(s).

For various reasons, this project failed.

But Professor Muller's intuition was accurate; he was German, and he must have studied Riemann's German manuscript.

Professor Muller had long suspected that Riemann's "unpublished expression" was not just one.

"Old ginger is spicier. It's becoming increasingly difficult for Professor Muller to complete high-intensity, big data calculations and derivations, but his strategic vision and mathematical intuition are exceptionally sharp." Shen Qi is now understanding Professor Muller more and more. Time will tell; only by spending a lot of time with someone can you discover their true strengths and weaknesses.

Muller's office.

Shen Qi wrote the underlying principles of the "twin matching method" and the first expression on the blackboard.

"My God!" Muller was shocked. He stared at the blackboard, speechless for a long time.

Shen Qi tapped the blackboard. "Professor Muller, what do you think of my new concept?"

The elderly Muller, suddenly seized by youthful exuberance, gestured wildly with great excitement. "With just this new 'twin matching method' and the first expression, Shen Qi, you are qualified to give a presentation at the International Congress of Mathematicians. IMU should give you at least 30 minutes of presentation time! This is a major breakthrough in the Riemann Hypothesis! If you can derive all the theoretically existing expressions, my God, the Riemann Hypothesis will be solved!"

Shen Qi calmly said, "Professor Muller, I need your help, I need the help of the entire team. Looking back on your and Mary's views a year ago, I realize how foolish and stubborn I was. Now, let the Muller warship set sail again. I hope that through the strength of the team and the proper allocation of resources within the department, we can finally complete the proof of RH."

Muller was both surprised and pleased. "Are you sure you want to do this, Qi? You are now fully capable of handling the RH certification work on your own. In a few months, I will no longer be your mentor, and you will be able to start your own research project."

Shen Qi smiled and said, "Allen, even if you weren't my mentor, we could still be partners. Thank you for taking care of me over the past year and providing me with a free academic environment, which has played a crucial role in my growth. Remember the 'Muller -Shen Theorem'? We will always be partners, and this theorem will always stand."