It can only be said that the two authors are really very patient.
They explain so of the special relationships, summation techniques, and other content so thoroughly, it’s practically a standard foolproof textbook.
It’s not an exaggeration to say that Qiao Yu thinks even soone completely unfamiliar with analytic number theory, after reading this book’s derivation steps and application examples, could be considered to have entered the field.
As he watched until dawn, Qiao Yu went out for breakfast and returned to his study. After stretching, he picked up a pen and began to sketch a frawork on a sheet of paper...
Since he wanted to use a new thod to solve the Twin Pri Conjecture, he couldn’t follow the old paths of others.
Qiao Yu planned to start with the pseudo-complete spaces he was most familiar with and the Langlands Program.
The Langlands Program aims to establish profound connections across different fields of mathematics, closely tied to symtry in number theory and representation theory.
Therefore, one could certainly consider directly viewing the nature of twin pris as issues of symtry and mapping in so kind of geotric or algebraic frawork.
These are obvious considerations.
The current problem is, to achieve this, he needs to construct a new category, with its objects naturally being pairs of twin pris.
Then define appropriate morphisms to express the structural relationships of these number pairs.
Next is constructing a topological structure.
Schulz’s pseudo-complete space theory embodies almost complete structures, which ans it can capture boundary behavior.
Coincidentally, the core of the Twin Pri Conjecture is about studying the limiting properties and boundary distribution of pri pairs.
This ans combining both, one can establish a pseudo-complete space for twin pri pairs.
Theoretically, all twin pri pairs can be mapped into this pseudo-complete space, allowing each pair to form an approximately equidistant sequence within this space.
Then, one can introduce topological tools to locate possible topological invariants in the relationships between twin pris.
Then directly define new algebraic and geotric objects, construct a twin pri cluster, and consider defining relationships between twin pri pairs through group or module structures.
Alternatively, one could establish a twin pri moduli space, mapping all twin pri pairs so that the geotric features of this space reflect the properties of twin pris.
This way, tools such as Hodge structures can be used to find the periodic regularities in the distribution of twin pri pairs...
Quickly, Qiao Yu filled the draft paper in front of him with content, using arrows and casually drawn figures to represent his thought path.
Of course, this is just a rough conceptual chart; what is useful and what is rely his imagination, Qiao Yu won’t know until he starts working on it.
However, there’s no rush for these tasks; Director Tian’s requirent is only for him to submit the project before school starts, which ans he just needs to complete a feasibility report.
Honestly, Qiao Yu feels that his advisor slightly underestimates him again.
As long as he doesn’t have to give a complete proof, for such purely theoretical project idea reports, he can write one to submit every day without repeating them.
Anyway, Qiao Yu feels that whether it’s solving the Twin Pri Conjecture or the Riemann Conjecture, tools capable of capturing finer number theory structures need to be introduced.
Traditional number theory research is mostly limited to algebraic thods, which is evidently outdated.
For more ticulous work, geotric mapping ideas must be introduced, directly transforming the arithtic properties of number theory into geotric space changes.
Of course, if other goals are to be achieved, it’s best to construct a computational tool.
Ideally, it not only verifies the distribution of zeros of L-functions but also checks directly within a finite number of evolutionary steps if the zeros lie on the critical line, maximizing supercomputing efficiency...
A plethora of ideas erged at once, and Qiao Yu recorded all of them.
He planned to discuss with Director Tian in the car when he went to pick up Qiao Xi in the afternoon, conveniently showing off his ambition.
Since he was going to do it, he decided not to aim for a more precise position. Professor Zhang had already derived up to 70 million digits, though that number is large.
But even if his thod could reduce the difference between infinitely many pri pairs to 6, so what?
K still can only equal 3 instead of 2, which does not constitute a full proof of this conjecture; it’s aningless!
At that mont, Qiao Yu suddenly had an epiphany—the thing about mathematics, isn’t it just a Creator’s ga?
It’s nothing difficult!
Although this statent seems to have a hint of arrogance right now, Qiao Yuzhen thought exactly that.
Yes, at this mont, Qiao Yu found that the Twin Pri Conjecture and even the Riemann Conjecture were not difficult; he just hadn’t figured out how to create yet.
Thus, as he thought and was pleased with himself, the scheduled eting ti quickly arrived, and Director Tian called him out with a phone call.
Director Tian had really prepared two cars, parked front and back at the Math Research Center’s entrance.
Director Tian sat in the back seat of the first car, rolled down the window, and waved to him: "Why are you standing there in a daze? Get in the car."
"Oh, coming,"
Qiao Yu responded, quickly walked to the other side, opened the car door, and sat down. As the car started moving, he couldn’t help but ask: "Director Tian, why do we need two cars? Isn’t one car that can seat more people enough?"
"Is it convenient to talk with everyone in one car?" Tian Yan Zhen glared at Qiao Yu and said.
"Oh, I see."
Qiao Yu thought and then said, "Actually, it’s not very inconvenient. You can boast about more in front of my mom, let others hear how excellent I am; it doesn’t seem bad, does it?"
Before Tian Yan Zhen could speak, the driver in the front seats suddenly couldn’t help but let out a laugh, though it was just one sound, he quickly stopped.
Qiao Yu glanced at the driver, thinking, how does this person seem both professional and unprofessional?
"Less nonsense! When I praise you, it’s only with reason and evidence!" Tian Yan Zhen glared at Qiao Yu and said.
"Oh, then let provide you with a reason to praise . I’ve already completed what you asked to think about during the sumr. Just regarding the Twin Pri Conjecture, I feel like I can directly solve this problem," Qiao Yu said confidently.
And then, in a very arrogant tone, he expressed his insight: "Mathematics is really very simple; it’s just a Creator’s ga!"
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