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Chapter 195 The Relics of the Mathematics Department (4.8k)

Because Göttingen is a university town, it becomes deserted around Christmas.

In the 60s, the area had a permanent population of around 12, of which 3 were students.

At least 70% of them are students from other places.

This year is different from previous years; Göttingen is more lively than usual.

Mathematicians from all over the world first fly to Frankfurt and then take a train to Göttingen.

If the city as a whole is a bit more lively, then the local newspapers in Göttingen and the regional television station in Lower Saxony, where Göttingen is located, can be described as exceptionally lively.

In 1965, West German television broadcasting consisted mainly of the Public Broadcasting Union (ARD) and the Second German Television (ZDF).

ARD consists of multiple regional broadcasters, each covering one or more federal states and providing national and regional programming.

NDR's third channel is specifically designed to serve regions such as Lower Saxony.

Every piece of news related to Lin Ran's return to Göttingen was reported as local news, down to the smallest detail.

They're practically ready to put up a banner that says, "Professor, welcome home."

"Fox, it's a good thing you contacted me in advance, otherwise it would be very difficult to find enough hotels in Göttingen, especially rooms like 223 and 227, which are being snapped up." Doilyn led the Columbia University group to the student dormitories at the University of Göttingen.

Fox, who led the team to Göttingen ahead of time, didn't even consider finding a hotel. He directly contacted Doiling and asked them to help arrange student accommodation.

Fox knows very well that doing research, especially when it involves breakthroughs, requires working day and night, and it's impossible to take a break after four hours like a regular lecture.

It's a 100% non-stop marathon.

Lin Ran kept watching, and as viewers, they certainly couldn't stop watching either.

We have to listen to it from beginning to end.

In this situation, the quality of the place to live doesn't matter; having a place to stay is the most important thing.

Fox thought the best place to settle down would be the student dormitories at the University of Göttingen.

As for two people sharing a room, that's not important at all.

He didn't even plan to go back to his dorm; Fox and the math professors he was leading each had a sleeping bag.

The plan was for everyone to just lay their sleeping bags on the ground and sleep there.

Having a student dormitory as a place to stay is simply a matter of being prepared for any eventuality.

When Fox heard Doyling say this, his mathematician's keen insight was still there: "Because Randolph is trying to prove the twin prime conjecture, these prime-numbered rooms are being snapped up, right?"

He immediately noticed the characteristic of these two numbers: they were both prime numbers.

Doyle said with a wry smile, "Not only that."

He told Fox about Lin Ran's actions in London and his ideas about the Prime Number Enlightenment at the Clarridge Hotel.

This was relayed to him by Siegel, who was still in London, and then Doilyn spread it as an anecdote.

It was reported by local media.

There were already few rooms that met the criteria, and with this particular time, rooms with numbers 257 and 523 were sold out. Then, rooms with the other 13 three-digit twin prime numbers that had the same digits were also snapped up.

After listening, Fox laughed and said, "I feel this will become a tradition among mathematicians in the future."

For example, at next year's International Congress of Mathematicians, when the organizers book hotels for everyone, everyone will definitely want to stay in rooms 257 and 523, followed by rooms 223 and 227.

Doyle said with a wry smile, "That's right, this is still based on the premise that the professor has not proven the twin prime conjecture."

If he succeeds this time, then everyone should have no doubt that the twin prime room is helpful for thinking.

Fox laughed and said, "Looks like the first thing I'll do when I get back to Columbia is change all the room numbers in the math department to prime numbers, so no one will argue about it."

However, such prime numbers are finite, and as time goes on, everyone's office door numbers will become increasingly longer.

Doiling said, "The professor's influence is too great. The local newspapers in Göttingen even joke that in Göttingen now, you could hit a whole group of mathematicians with a single brick."

On the afternoon of January 4, 1965, Göttingen train station was packed with people.

Lin Ran traveled from London to Göttingen by train, making two transfers. He was accompanied by Siegel, Jenny, and high-ranking West German officials, with security personnel leading the way and following behind.

Police could be seen everywhere at the train station.

Security at Göttingen train station has never been so comprehensive.

He was greeted by Otto Kummer, the rector of the University of Göttingen, the head of the mathematics department, Doiling, and several senior professors.

Outside the station, student volunteers held up welcome signs, while journalists from all over West Germany and even Europe gathered, notebooks in hand, to record this historic moment.

“Professor, I look forward to witnessing your miracle,” Otto said after shaking hands.

Doilyn continued, "Professor, the stage is set up, and everyone is waiting to see your performance. The whole of Göttingen is eagerly anticipating it."

The lecture was held in the main auditorium of the University of Göttingen, an 18th-century classical building known for its dome and carved columns, which can accommodate 500 people.

According to the history of the University of Göttingen, the Great Auditorium is often used for important academic events, such as lectures by Nobel laureates.

On January 5, 1965, the auditorium was packed, with extra audience members crowding the corridors. The university set up loudspeakers in nearby classrooms to broadcast the event and arranged temporary seats in the courtyard for students and scholars who could not attend.

In addition to these, the local television station in Göttingen set up cameras, intending to broadcast the entire event live.

Inside the auditorium, the center of the stage was covered with blackboards, nothing but blackboards.

"Ladies and gentlemen, let us first give a warm round of applause to welcome Randolph Lin back to Göttingen," Otto said. "Göttingen is the professor's alma mater, and we are proud and honored to have nurtured such an outstanding student as Randolph Lin. Now, let me hand over the reins to Randolph."

Lin Ran whispered to Siegel, "Professor, I'm leaving the recording to you."

Siegel nodded. "No problem."

Lin Ran walked onto the stage, and thunderous applause erupted from the audience.

After the applause subsided, Lin Ran said:
"Ladies and gentlemen, respected colleagues, dear friends, good morning!"

I feel incredibly honored to be back in Göttingen, the land that nurtured my mathematical dreams. Standing in this grand hall, I feel as if I've returned to my student days, when I listened to Hilbert's successors lecture on number theory, stayed up all night studying Euclid's proofs, and tried to unravel the mysteries of prime numbers.

Of course, back then I never imagined that I could prove Fermat's Last Theorem, propose the Randolph Program, or that one day I would stand here trying to challenge the twin prime conjecture.

It has been exactly 65 years since Professor Hilbert posed this problem as the eighth problem in his report at the International Congress of Mathematicians in 1900.

Lin Ran turned around and wrote "3, 5", "5, 7", and "11, 13" on the blackboard. Then he turned back, his gaze sweeping over the audience, and his tone became solemn.

"You all recognize these numbers."

They are twin primes, a pair of primes that differ by 2.

They seem simple, but they conceal the speculations of our predecessors: Are there an infinite number of such pairs?

This problem can be traced back to ancient Greece. Euclid proved the infinity of prime numbers, but he left us with an unsolved mystery regarding twin primes.

Fast forward to the 19th century, and mathematicians began to seriously consider this problem.

In 1849, Alphonse de Polignac proposed a more general conjecture, asserting that for any even number k, there exist infinitely many pairs of primes p and p′ such that p′p=kp'-p=kp′p=k.

When k=2, this is our twin prime conjecture.

Lin Ran then wrote p′p=2p'-p=2p′p=2 on the blackboard.
"This conjecture seems intuitive, as number theory always is; it's very intuitive, and everyone can understand the problem. But in the rigorous world of mathematics, it's like a mountain that's hard to climb."

Lin Ran spoke very quickly in English, and his standard English ensured that every scholar present could hear him clearly.

Germans are not as committed to the German language as the French.

Lin Ran fell into deep thought, slowing his pace, placing his hands behind his back, and gazing into the depths of the auditorium, as if tracing back history. "At the beginning of the 20th century, mathematicians began to use more powerful tools to tackle the problem of prime number distribution. In 1919, Norwegian mathematician Viggo Bren made a breakthrough."

He invented a technique called the sieve of Brun, which proved that the sum of the reciprocals of twin primes is convergent.

Lin Ran then wrote on the blackboard:

What does this mean? Compared to the divergent reciprocals of all primes, twin primes are so sparse that the sum of their reciprocals does not even tend toward infinity.

Brun's theorem tells us that twin primes are not as common as ordinary primes. Their sparsity makes proving infinity exceptionally difficult. But isn't that the charm of mathematics? Our creativity is truly unleashed when faced with a seemingly impossible problem. ********Dolph walked to the side of the podium, picked up a glass of water, took a small sip, and glanced at the audience.

The reporters whispered among themselves in a corner, trying to catch every word Lin Ran said.

The atmosphere in the auditorium shifted from tension to anticipation, as the audience was drawn into the world of prime numbers by his narration.

“While Brun’s work did not prove the conjecture, it pointed us in the right direction. Hardy and Littlewood later provided heuristic support using the circle method, estimating the number of twin prime pairs to be approximately (log)2C(logx)2x, where is the twin prime constant, which is about 1.32032.”

Lin Ran then wrote the formula on the blackboard.

"But these are all probabilistic predictions, and are still far from being truly proven."

Today, I stand here not to repeat these predictions, but to show you a possible answer—a proof combining analytic number theory and sieve methods, attempting to unveil the mystery of the twin prime conjecture.

Over the next six days, we will embark on this journey together.

From the distribution of prime numbers to the ingenuity of the sieve method, and then to the profound tools of analytic number theory, I hope to convince you that this conjecture is no longer a conjecture, but a theorem.

Of course, I know that many of you, especially the professors in Göttingen, will scrutinize my proof with the most rigorous standards.

This is exactly what I've been waiting for! Let's get started!

The audience below the stage were applauding, and Siegel was too, but he had a different feeling than the others.

Professor Siegel was certain that this was Lin Ran making up for the graduation thesis defense he had been unable to do at the University of Göttingen.

He sat up straight, thinking, "Randolph, let me witness your legend and prove with my actions that the Göttingen School has not perished, but will become even more glorious because of you."

Lin Ran turned around and wrote Day 1 on the blackboard.

From the moment they wrote Day 1, the scholars present felt a sense of rapid progress.

Because Lin Ran was too fast.

Lin Ran first needs to produce Zhang Yitang's result, which is that there are infinitely many pairs of prime numbers whose difference is less than 7000 million. Then he needs to produce Tao Zhexuan's improved result, which reduces this difference from 7000 million to 246.
However, he could not directly use Zhang Yitang's results.

Because Zhang Yitang's paper is based on the GPY sieve method and the 4/7 level results of Bombieri, Friedlander and Iwaniec on the distribution of prime arithmetic series.

The GPY sieve method only appeared on arXiv in 2005, while the paper by Bombieri, Friedlander, and Iwaniec appeared in 1987.

Lin Ran needed to reproduce the results in 1965, but he couldn't just use Zhang Yitang's results directly; he had to write the prefix paper first.

Therefore, the first day

The formulas on the blackboard piled up one after another. Lin Ran spoke very little but wrote a lot, and kept walking back and forth.

Once the blackboard is full, push it to the side.

They would write on each sheet and then push it away; the Göttingen University had prepared a portable blackboard beforehand.

The University of Göttingen was happy with this arrangement; they didn't want to erase a single one.

If Lin Ran can truly prove it successfully, these will be sacred relics of the mathematics department, becoming more valuable the longer they are passed down.

"Okay, I've figured out my core ideas."

I start with acceptable k-tuples.

These k-tuples, where each prime number p has at least one residue class that is not covered, ensure that they are likely all prime numbers.

My goal is to prove that there exists a k such that there are infinitely many n such that the tuple ({n+h_1, n+h_2, ..., n+h_k}) contains at least two prime numbers. This would imply that the gaps between prime pairs are finite.

I used a variant of the Selberg sieve to construct a weight function to detect cases where there are at least two prime numbers in a tuple.

By optimizing the parameters, I estimated the number of n that satisfy the conditions. The key is to ensure that the main term is greater than the error term.

"Controlling the error term requires knowledge of the distribution of prime numbers in the arithmetic series."

We must first allow the average modulus to be up to x^{1/2}.

Then it is enhanced to suit smoothing modulus and expand distribution levels. This step is to enable the sieve method to handle large k values.

Using these tools, I have proven that for sufficiently large k, there exist finite N such that there are infinitely many pairs of prime numbers whose difference does not exceed N.

Then we first find an N, and then gradually reduce the value of this N until it eventually equals 2.

After Lin Ran finished speaking, the scholars in the audience looked very serious.

Because Lin Ran's proposed approach is not a strange one; it is very orthodox and not fundamentally different from the thinking of mathematicians around this problem in the past.

However, the method Lin Ran mentioned will have some innovative aspects.

This approach alone is clearly insufficient to solve the twin prime conjecture.

"We'll start with the first step, beginning with analytic number theory. We'll work backward from Mark Barban's results."

First, we need to prove that for a specific Q near x, if we ignore the logarithmic term, the average error can be as small as half of x.

Then, this result is extended by increasing the modulus from one-half to four-sevenths, so that the error term of the prime number distribution holds true with a larger modulus, which is applicable to the sieve problem in analytic number theory.

Lin Ran started writing quietly, only speaking when explaining.

He said very little.

As they wrote, the mathematics professors from Princeton in the audience became numb.

Because the result Lin Ran casually wrote down became a major achievement that the Institute for Advanced Mathematics at Princeton will publish this year.

Taking half of x is called the Bumbery-Vinogradov theorem in mathematics; also known as the Bumbery theorem, it is a major achievement in analytic number theory and is related to the distribution of prime numbers in an arithmetic sequence that takes the average of a series of moduli.

The first such result was obtained by Mark Barban in 1961, and the Bumberly-Vinogradov theorem is a refinement of Barban's result.

This finding was published in 1965 and solved by Enrico Bomberi and Ascord Vinogradov at Princeton, hence the name Bomberi-Vinogradov theorem.

It wasn't until 1987, more than 20 years later, that they were able to advance this result from one-half to four-sevenths.

Lin Ran is now tasked with verifying their results on the spot, and then surpassing them in the process.

The more Lin Ran wrote, the darker the faces of the professors from Princeton became.

Since Lin Ran's writing on the one-half result was impeccable, it means that his extrapolation to four-sevenths is also highly likely to be correct.

This feeling of frustration is like you painstakingly leaping and dodging, using all sorts of maneuvers and unleashing powerful attacks to finally defeat a monster, only to have it instantly killed by someone else with a single basic attack.

They can hit faster than you, and their fighting style is even more graceful.

"Okay, as you can see, we have completed the proof here."

We have just proven that the distribution of prime numbers in an arithmetic series can reach the level of 4/7.

Specifically, it shows that for modulo ≤ 4/7, the distribution error term of prime numbers in the arithmetic series (mod) (gcd(,) = 1) can be effectively controlled.

This result expands the modulus range, making the sieve method applicable to a wider range.

The main idea here is to overcome the limitations of traditional methods and improve the analytical capabilities of prime number distribution by introducing bilinear form estimation and decentralization techniques.

We have laid the foundation for the finite gaps in the overall approach to the twin prime conjecture.

(End of this chapter)