A genius? I just love studying.

Who is the author of this paper in Chapter 268?

He excitedly burst into Fefferman's office and shouted before Helmut realized that Fefferman wasn't alone in the room.

Everyone here is a renowned international mathematician, and we all know each other.

"Oh, Professor Zhang, Gus, Maynard, you're here too."

Helmut greeted the group, "What a coincidence!"

He placed the paper he was holding on Fefferman's desk, then took out his phone. "I sent you an email. Print out the attachment. Everyone can take a look."

Fefferman looked at the rambling Helmut with a puzzled expression, but still turned on his computer and went to his email.

Proof of the Twin Prime Conjecture

The title of the attached PDF was simple and unpretentious, but it certainly made Fefferman's eyes widen.

He also subconsciously glanced at Zhang Yitang beside him.

Zhang Yitang was able to rise from a dishwasher to a renowned mathematician by proposing a possible proof of the twin prime conjecture.

Surprisingly, someone has now proven the twin prime conjecture.

Upon opening the attachment, Fefferman regretfully did not see the author's name.

After quickly printing out three copies, he eagerly picked up the paper Helmut had brought and began to study it with great enthusiasm.

"?"

Zhang Yitang and his two companions stared blankly at the scene, realizing that Fefferman's behavior was undoubtedly extremely rude.

They knew Fefferman wasn't that kind of person, so the only explanation was that there was something fishy about the paper Helmut brought!
Everyone looked toward the printer that was creaking and groaning as it worked.

"As the elder, I should be the one to read the first copy. Do you two have any objections?"

Zhang Yitang strode to the printer, smiling as he looked at Gus and Maynard.

Gussman and the other man exchanged a glance, then walked straight behind Fefferman, their eyes glancing over Fefferman's shoulder at the paper.

By this time, Fefferman had finished reading the first page and placed it on his right.

Proof of the Twin Prime Conjecture

Upon seeing the title of the paper, Gus and his companion's pupils contracted, and they looked at each other with shock in their eyes.

They didn't speak, but their gazes toward Fefferman's paper on the table became much more serious, as if facing a formidable enemy.

At this moment, Zhang Yitang finally picked up the first paper he had just printed, and he saw the title of the paper: "Proof of the Twin Prime Conjecture"!
Taking a deep breath, Zhang Yitang was still slightly excited.
His improved sieve method, the bounded distance method, is now the most powerful tool for proving the twin prime conjecture. He proved that there are infinitely many pairs of primes with a difference of less than 7000 million. Soon, Terence Tao changed this number to 246. As long as this distance is changed to 2, the twin prime conjecture can be proven.

Someone has now submitted a proof of twin primes, which most likely used his bounded distance method.

Although he was not the one who proved it, his contribution cannot be erased. In the future, when people mention the twin prime theorem, Zhang Yitang will be an unavoidable figure.

After daydreaming for a while, he finally picked up the paper and began to study it.

Not long after, he frowned.

The authors of this paper reconstructed the sieve method. Although they also cited his bounded distance method, their proofs were ultimately very different. As it turns out, although Zhang Yitang's method seemed to be close to the truth, it was just a mirage and could never reach the final answer.

His method is wrong!
Zhang Yitang continued watching, unwilling to give up.

After reconstructing the sieve method, he transformed the zero-point term ∑ρx^ρ/ρ in the Riemann formula into a tool to control the oscillation of the prime number distribution, explicitly linking the ζ zero-point density with the sieve method error.

He tamed the zero-point oscillation!

In the final processing of the principal and remainder terms, the authors of this paper introduced a bilinear form estimation.
1. When d and e are coprime, the large sieve inequality is used for control;
2. When d and e are not coprime, the error can be compressed to: Ex(lnx)A (A>0) by utilizing the oscillation suppression property of λd.
Combining the main term and the remainder term, we can conclude that when x is large enough, there are infinitely many pairs of prime numbers (p, p+2)!

He actually proved the twin prime conjecture!

Zhang Yitang was shocked, yet still somewhat incredulous. He turned back to the first page and went through the proof process in the paper again.

There are no vulnerabilities.

Three hours later, Zhang Yitang had to admit the fact; at least, he couldn't find any flaws in the proof.

He had no choice but to look up at Fefferman and the other two.

The three of them looked up at him, and the four of them exchanged glances, already understanding the shock in each other's hearts.

This proof is correct!

They were certainly not shocked by the proof of the twin prime conjecture. What shocked them was that the author of this paper was clearly not interested in the twin prime conjecture. This was obviously just an unexpected bonus on his way to proving the Riemann Hypothesis.

Although he failed to tame Riemann's zero-point dragon, he glimpsed the galaxy composed of twin primes through the gaps in its scales!
If anyone else had claimed to be proving the Riemann Hypothesis, the four would have scoffed. But with this paper as a precedent, they suddenly felt that the person might actually be able to complete this impossible proof.

Although the twin prime conjecture is quite different from the Riemann Hypothesis, the method for proving twin primes may well be the key to proving the Riemann Hypothesis.

"Do you also think this proof is valid?"

Helmut had been waiting for a long time, and seeing the expressions on the four people's faces, he naturally understood what was meant.

Fefferman did not answer his question, but instead asked, "Who is the author of this paper?"

This is also what puzzles them greatly.

Generally speaking, they know internationally renowned mathematicians and are aware of the methods and research questions these peers are using. At their level, double-blind peer review is practically meaningless; they can guess whose paper it is just by looking at it.

For example, Guss's research focuses on large value estimation and zero density improvement of Dirichlet polynomials. He is skilled in metric geometry and harmonic analysis. If there are top papers in related fields, people will think of him first.

For example, Maynard, a top expert in analytic number theory, independently proposed an improved version of the sieve method before Terence Tao made his move, proving that there are infinitely many pairs of prime numbers with a gap of less than 600, breaking Zhang Yitang's record of 7000 million gaps, and the method is simpler and more efficient.

Furthermore, in recent years, they proved the Duffin-Schaeffer conjecture, thus improving the application of Khintchine's theorem in Diophantine approximation.

Zhang Yitang has recently been studying the Landau-Siegel zero-point conjecture...

The research areas of these famous mathematicians are not a secret; everyone knows it.

However, they did not know who wrote this paper.

This paper uses sieve analysis and harmonic analysis. If Zhang Yitang, Gus, and Maynard hadn't been there, he might have suspected one of the three.

But now it seems that it was clearly not them.

Zhang Yitang and the other two also looked at Helmut with curiosity.

“Of course it’s Professor Chen!” Helmut looked at Fefferman, completely bewildered. “Didn’t Professor Chen tell you a long time ago that he was going to study the Riemann Hypothesis?”

"?"

"Chen Hui?"

Fefferman's lips parted slightly, a look of disbelief flashing in his eyes, and he froze in his chair.

Chen Hui did say he wanted to study the Riemann Hypothesis, but how much time has passed?
It's been less than two months at most!

Gus and Maynard looked at Zhang Yitang with strange expressions.

They remembered that Zhang Yitang had just said that Chen Hui was doing this only to confuse S.H.I.E.L.D.

If Chen Hui proved the twin prime conjecture simply to confuse the public, then that would be even more terrifying.

Zhang Yitang's face flushed slightly; he was a little embarrassed.

Fortunately, Gus and his companion didn't look at it too much. Although they were focused on mathematics, they weren't idiots without emotional intelligence.

"Seeing is believing, indeed!"

Gus suddenly remarked, "Professor Fefferman, could you arrange a seminar for us before the lecture? We would like to have a face-to-face exchange with Professor Chen."

"of course."

"If Professor Chen has no objections."

Fefferman readily agreed, as he also wanted to see how far Chen Hui had progressed in his research on the Riemann Hypothesis.

……

A week passed in the blink of an eye. In Chen Hui's office,

"The key to verifying the Riemann Hypothesis is to prove that all non-trivial zeros lie on this line."

Now, suppose I tell you that, through some method, we have reduced the upper bound σ of the real part of the zero to 0.41. How would you improve this result?

Chen Hui's voice rang out in the office.

Elena and Deng Leyan stared blankly, somewhat relieved that they hadn't chosen the analytic number theory path.

They started with questions at the level of the Riemann Hypothesis, which is something an undergraduate student like them should be able to answer?

"Teacher, you've reduced the upper limit σ of the real part of the zero to 0.41?"

Michael was ecstatic.

These days he has studied a lot of papers on the Riemann Hypothesis, not just the pile Chen Hui gave him, but also some reference materials he found himself.

He knew that Ingham's theory of zero-point density upper bound, proposed in 1940, reduced the upper limit of the real part of the key parameter σ to 0.6. For more than eighty years there was no progress until a few years ago when Guth and Maynard reduced this parameter to 0.52.

This has already earned high praise from Terence Tao.

If the teacher were to reduce this parameter to 0.41, one can imagine the sensation it would cause in the mathematics community.

But then he thought about it again and realized that compared to his teacher's previous achievements, this seemed rather ordinary, but he was still very excited.

"answer my question."

Chen Hui remained unfazed. He had indeed reduced σ to 0.41 by improving Gus and Maynard's method, but this was already the limit of that method. It could only be considered as eliminating an incorrect path for him, and could not be considered a significant achievement.

Michael became even more certain of his guess, his face beaming with excitement.

But he quickly regained his composure, and the papers he had been studying these past few days automatically came to mind. Although he kept saying he wanted to have a party, he hadn't been idle this week. He not only finished studying the stack of papers Chen Hui had given him, but also looked up a lot of information on his own.

Just a few minutes later, he had a preliminary answer: "It might require analyzing the large value estimate of the Dirichlet polynomial. The traditional method uses the moment estimate, but if we can find a better exponent and estimate, or use Fourier analysis of the automorphic form..."

"Stop." Chen Hui raised his hand. "What exactly did you mean by 'moment estimation'?"

Michael paused for a moment, "For example, calculate the average modulus of ζ(1/2+it) raised to the power of 2k, and estimate its relationship with the zero density through integration."

Why is moment estimation effective here?

“Because…” Michael’s voice lowered, “the zero-point density is related to the oscillation frequency of the zeta function near the critical line, and higher-order moments can capture more refined oscillation patterns.”

Chen Hui turned around, pulled out a piece of draft paper from the drawer, and quickly wrote down a series of formulas on it. "This is the contraction inequality that Guth and Maynard used last year, which reduces the high-dimensional Kakeya problem to a plane. If you were to apply it to the estimation of Dirichlet polynomials, how would you adjust it?"

Michael's pupils contracted slightly. He had seen this formula when he was researching, but at the time he thought it was too abstract.

Michael's pen moved across the draft paper, overlapping the symbols of complex analysis with the framework of geometric analysis. "Suppose we have a function f(t) = ζ(1/2 + it) in t. To estimate its maximum modulus in the interval [T, 2T], the traditional method uses the L^p norm. But if we use decomposition techniques to break f(t) down into low-frequency and high-frequency components..."

“That’s enough,” Chen Hui interrupted him. “Your derivation just now overlooked a problem: the phase of the Dirichlet polynomial is linear, while the phase of the Kakeya problem is polynomial. This difference will change the applicable conditions of the contraction inequality. If you apply it directly, it will cause the error term to explode.”

A brief silence fell over the office.

Michael looked down at his notes, then suddenly looked up. "Professor Chen, you mentioned in your seminar last week that the breakthrough in the twin prime conjecture depends on 'a more uniform distribution of primes in an arithmetic series.' Is this related to the distribution of zeros in the Riemann Hypothesis?"

Chen Hui's eyes lit up. "A very good question."

The twin prime conjecture concerns the interval between primes, while the Riemann Hypothesis concerns the error in the prime theorem.

In simple terms, if the Riemann Hypothesis holds true, the error term of the prime number theorem is O(x^(1/2)logx), which means that prime numbers are more evenly distributed among natural numbers than any 'reasonable guess'.

The lower bound of the interval between twin primes is essentially a manifestation of this uniformity—if the distribution of prime numbers is uniform enough, they will not disappear for a long time, but will only appear in pairs occasionally…

The two men were exchanging questions and answers in the office, but Elena and Deng Leyan, who were listening in, suddenly felt a sense of urgency. They had originally only regarded each other as rivals and thought that this new black guy was probably the teacher's joker.

But now it seems that this guy's strength is terrifying!
"very good!"

The two-hour Q&A session made Chen Hui's eyes shine brighter and brighter. This world is indeed never short of geniuses, and geniuses are often attracted to each other.

Michael let out a long breath, and once he had passed the teacher's test, he flashed a bright smile, revealing his pearly white teeth.

"Teacher, the Riemann Hypothesis is a huge mountain. I think we can start with smaller problems first."

Michael quickly regained his cynicism, saying, "We can start by proving something simpler, like the twin prime conjecture or Goldbach's conjecture."

Elena and Deng Leyan exchanged a glance, both seeing the smile in each other's eyes. Then she turned around, went to her desk, picked up the latest issue of the "Annals of Mathematics," and handed it to Michael.

The cover of this issue of the "Annals of Mathematics" prominently features the proof of the twin prime conjecture, and the author, of course, is Chen Hui!
(End of this chapter)