Technology invades the modern world

Chapter 196 Day 3 (5k)

After Lin Ran finished speaking, the number theory experts present all stood up and applauded.

Other doctoral students or mathematics professors in other subfields, who were not familiar with the topic, also politely applauded.

For a moment, the reporters present were completely baffled.

After their applause subsided, he found a professor who had stood up and asked in a low voice:
"I'd like to ask, is this really that amazing?"

"That's amazing. The professor has found another cornerstone for analytic number theory. The two results he presented today can be used not only for the twin prime conjecture, but also for other number theory problems."

The professor not only found the modulus, but also expanded the range of moduli. The increase in the range of moduli directly enhanced the capabilities of sieve methods and distribution analysis.

Furthermore, the bilinear model and decentralization techniques he used provide us with new tools for analytic number theory and sieve methods.

In short, this is already a very impressive achievement.

Most mathematicians could win the Fields Medal with this result alone; the trip to Göttingen would be worthwhile just for this result.

恩里科·邦别里正是靠这个成果拿的1974年的菲尔兹，他把模数范围从1/2扩展到4/7，对标准定理做重大改进都要到1987。

Lin Ran's current content is equivalent to at least two Fields.

However, this is only the beginning.

The mathematicians present, especially those working in number theory, must be thrilled.

"Okay, everyone, we've now advanced the modulus to four-sevenths. Sorry, time is tight, so I won't go into the discussion."

If you have any questions, please write them down and I will try my best to answer them. If I don't have time this time, I will answer questions again at Columbia University when I return to New York.

Fox shouted from below the stage, "Okay, no problem, Professor, you can continue."

Doilyn was speechless. What's with all the yelling? This isn't your home turf. Are all Americans so annoying?
However, considering that this was an unprecedented occasion and time, he did not lose his temper.

"We need to move forward now."

Lin Ran wrote a new formula on the blackboard:
Sixty years later, this formula became known as the Elliott-Halberstam conjecture. The EH conjecture was proposed by Elliott and Halberstam in 1968 and published in Symposia Mathematica. As of 2025, the conjecture has not been proven.

To put it another way, if this conjecture is proven, it means that the distribution error of prime numbers in arithmetic series with a modulus ≤ 1 can be effectively controlled, far exceeding half of the error in the standard theorem.

The twin prime K=246 can be quickly advanced to K=6, which is almost one step away from K=2 required for the twin prime conjecture.

Nathalie Debouzy's 2019 work, for example, improved the asymptotic sieve method and, assuming the EH conjecture holds, there are infinitely many near-twin primes. What is a near-twin prime? It means that if p is a prime number, p-2 is a prime number or a semi-prime number.

The EH conjecture is so important that later mathematicians have even begun to assume that it is true.

In other words, Lin Ran can no longer rely on the wisdom of future generations and must rely entirely on himself to eliminate the EH conjecture first.

It could even be said that the EH conjecture is a conjecture that the modulus is infinitely close to 1, and if we want to push the EH conjecture further back, that is, directly to 1, we need a completely new mathematical framework.

Therefore, after entering this stage, Lin Ran's speed obviously slowed down.

The more critical issue is that Lin Ran cannot directly use the existing theorems or lemmas from sixty years later. All the tools needed from sixty years later must be rebuilt on the spot in the auditorium of the University of Göttingen.

“Bilinear form and decentralization won’t work; this can only advance to four-sevenths at most.”

"Type II estimation, relying on short interval distribution control and smoothing modulus optimization, is also not enough; it still cannot be pushed to this extent."

"The zero-point relation of the L function will be a path."

The EH conjecture involves an error term for the average modulus q, where each q corresponds to a Dirichlet character χ (mod q), and the zero-point influence distribution of its L function.

The proof of the Bombieri-Vinogradov theorem relies on the zero density estimation, which controls the number of zeros of the L function around Re(s)≈1.

The EH conjecture requires stronger zero-point control, which involves the distribution of zeros within the critical band. Then, it relies on the indirect support of the GRH (Gross Principle).

Lin Ran wrote on the blackboard and erased it, then wrote it again.

The scholars present are well aware that this issue is very important.

The hypothesis itself is valuable enough.

It continued until 11 p.m., when Lin Ran began to speed up the pace of his chalk writing, without pausing for a moment.

The students who were responsible for changing the blackboard next to him had changed twice.

He wrote on all thirty blackboards without stopping for a moment.

There were only about twenty professors sitting in the audience, and many more people were lying on the ground in sleeping bags.

As the sound of chalk rubbing against the blackboard became clearer and faster, those who were awake woke up those who were asleep.

Everyone, pay attention to what's on the blackboard.

"This is?"

“That’s right, Randolph has found a way out.”

"We are indeed witnessing history. The twin prime conjecture is just the final destination. We are now enjoying the scenery along the way to the destination."

"I just fell asleep. Which path did Randolph choose?"

"I think it involves extending the difference distribution describing the zeros of the zeta function to the Dirichlet L function to influence the average behavior of the arithmetic series. If the zero distribution conforms to the stochastic matrix model, then it means that it can support his conjecture on error control."

"This is an idea, but whether it's feasible depends on its specific design."

After finishing writing, Lin Ran looked at the result in front of him and felt a sincere sense of accomplishment:
"Okay, that's enough for today."

As you can see, I'm so sleepy.

The current results deepen our understanding of the distribution of prime numbers and provide a preliminary tool for proving the twin prime conjecture.

Its breakthrough lies in surpassing the limitations of previous moduli.

In proving this conjecture, I analyzed the nontrivial distribution of the zeros of the Dirichlet L function.

By assuming that the zeros are sufficiently sparse within the critical band, the average behavior of the error term is estimated. Then, a novel sieve method is designed, combining bilinear form estimation and decentralization techniques to optimize the modular decomposition, thus overcoming the bottleneck of traditional methods.

Finally, a new lemma is used to control the high-dimensional exponent sum, ensuring that the error term satisfies the conjecture requirements.

Lin Ran made some final annotations on the blackboard.

"Everyone, I'm going to sleep now. I expect to continue in six hours."

Lin Ran didn't leave; he went directly to a small room next to the auditorium to rest.

The professors and doctoral students below the stage had already crowded to the front to see what was on the blackboard.

Lin Ran wrote on a total of thirty blackboards throughout the day.

The Bumbery-Vinogradov theorem and its enhanced form are easy to understand.

Moreover, Princeton had already formulated the Bombelli-Vinogradov theorem, so they quickly grasped both the Bombelli-Vinogradov theorem and its enhanced form.

Then came the EH conjecture.

Since the EH conjecture itself didn't even exist at that time, Lin Ran essentially handled everything from proposing the conjecture to proving it himself.

"It's so beautiful, it's practically a work of art."

"This is the result of a super enhancement."

Is there any room for simplification here?

"No, zero-point density estimation and pairwise correlation conjecture might simplify the professor's proof of this conjecture, but we still need to think about it carefully."

"The approach of controlling the high-dimensional exponent to ensure that the error term can meet the conjecture requirements is too ingenious."

"No, I have to go back quickly and send today's results to my colleagues who are still at school."

Using telegraph for mathematical papers is impractical. Theoretically, mathematical papers could be simplified into plain text, encoded in ASCII or Baudot characters, and sent in segments via telex, but in practice, it would be extremely difficult to express them accurately.

Nowadays, people usually use a fax machine to scan papers and send the images directly.

Due to the high cost of faxing, even though Göttingen is a university town, there are only a handful of fax machines.

However, Lin Ran's results today were so astonishing that both the results themselves and the methods he used excited the number theorists, who wanted to share them with their colleagues at the university as soon as possible and urge them to come to Göttingen to witness the miracle.

Vacation? What's the point of taking a vacation at a time like this? Witnessing a miracle in Göttingen is the most important thing.

Even though it was late at night, the scholars present, whether they had rested or not, were now full of energy.

Considering that Lin Ran had already gone to the adjacent lounge to rest, they lowered their voices to discuss today's results.

"In any case, the fact that we were able to advance the modular system to this extent is already a remarkable achievement."

"Remarkable? It's at least one of the most important achievements in number theory this century," Doilyn corrected.

"Professor Siegel, no, no, it's not time to draw a final conclusion yet. There are still five days. Judging from today's results, Randolph may be able to complete the twin prime conjecture."

In a sense, the twin prime conjecture is comparable to the Goldbach conjecture.

If he can complete it, then it will undoubtedly be the most important achievement in number theory this century, and we can remove it as one of the most important achievements.

"Unless someone is able to prove Goldbach's Conjecture in the remaining thirty-five years of this century," said Jean-Pierre, a top mathematician from France (who won the Fields Medal at age 27 and is one of the three major mathematics awards).

He had previously worked on commutative algebra and algebraic topology, but in the last decade or so, he has switched to collaborating with Grothendieck on algebraic geometry.

He continued, "I used to think about number theory too simply. There are too many analytical tools that can be applied to the field of number theory."

I think our current number theorists are far behind Randolph in their grasp of analysis.

Siegel smiled wryly and said, "We can't use Randolph as a standard to judge young students; that would be going too far."

Pierre shook his head and said, "No, I'm not saying that we should use Randolph's solid foundation in analysis to demand the same of young PhDs. Rather, we can't relax our requirements for the analytical abilities of young scholars just because their research topic is number theory."

Those who work in number theory should have analytical abilities comparable to those who work in analysis, and vice versa. Since Randolph proposed the Randolph Program, we have increasingly realized that there are very close connections between different subfields of mathematics.

We should encourage young scholars to excel in every field; we must cultivate well-rounded scholars.

Siegel said, "You plan to push for this standard to be applied to mathematics students at Parisian universities after you return to the École Normale Supérieure in Paris."

"Aren't we going back again?"

When Siegel says "going back," he means that in the early days, everyone trained all-around mathematics PhDs. Later, as modern mathematics became more and more abstract and people's energy was limited, students were no longer required to be experts in everything, because most people could not do that.

“That’s right, give it a try. At least we should hold the most talented young people to this standard.” Pierre nodded, his gaze still fixed on the densely packed formulas on the blackboard.

Siegel was older than Pierre, and he fully understood what the other was trying to say: "Are you saying that mathematics is changing, and our concept of talent cultivation also needs to change?"

Pierre nodded: "That's right. Randolph's actions and results have revealed the connections between different branches of mathematics, and these connections are becoming increasingly apparent. This is what we both thought during our discussions with Grothendieck."

It may be Randolph, or it may be someone else, but someone will always be able to turn the ideas proposed by Randolph's program into reality.

As modern mathematics is about to enter a new stage, we should also promptly change our strategies for cultivating young mathematicians.

Furthermore, my personal motive is that I also want to discover talents like Randolph for France.

"No audience wants to watch a TV program like this. The professor hardly speaks, and when he does, it's all very professional mathematical knowledge. Not to mention me, even among the math PhDs and professors in the audience, very few could fully understand what the professor was saying."

"Our viewers want more information; our current live stream is completely wrong," said program director Hermann Schmidt.
Late that night, not only were the mathematicians unable to sleep, but the NDR staff were also unable to sleep.

Aside from having many viewers at the beginning of the live stream, the number of viewers gradually decreased.

Because people can't understand it.

The NDR Hanover office meeting room was filled with smoke, with documents, ashtrays, and coffee cups scattered on the wooden table, and the NDR logo and program hanging on the wall.

Herman continued, "We need to solve this problem."

The professor's lecture had only been broadcast for one day when audience complaints piled up on the table.

Some people said it was an alien language, and others threatened to change the channel.

We can't just sit idly by; what about the six days of live streaming?

Herman is going bald.

I thought this would be an unprecedentedly popular live stream, and it was indeed popular, but only for the first half hour.

Half an hour later, it was all complaints.

"The phone hasn't stopped ringing since the broadcast started. Viewers say they want to see professors speak, but not lectures filled with formulas."

Some people even asked us if we had tuned into the wrong channel!
We have to take action, or we'll lose our audience.

Could we speak with the professor and have him share some interesting anecdotes from the moon landing? Could he talk about space travel and the moon? Now, we're tackling the twin prime conjecture live. I admit this is meaningful, and mathematicians worldwide are paying attention, but have we overestimated the audience's receptiveness?

Director Weber arrived at the office half an hour earlier. He said helplessly, "Who will ask the professor to adjust the content?"

Are you going? Do you have the face to do so?
It's clear that what the professor needs to do now is create a miracle in Göttingen. The person who can change his mind is in the White House. You need to find President Lyndon Johnson first, and then let him negotiate with the professor.

None of us can do that.

Moreover, this is a historic event, and the NDR has a responsibility to broadcast such an academic event live. We cannot give up just because of the opinions of the audience.

This way we can pre-record some commentary segments and broadcast them before and after each day's live stream to explain basic concepts. After the professor finishes speaking, we will contact the Berlin side and ask them to send a mathematics professor to provide support.

We need people who are more professional than us, but who can also express themselves in simpler language than the professors themselves, to explain things to the audience.

In short, Hermann, remember, NDR is not an entertainment channel; we are a public broadcaster, and education is our core mission. Randolph's lectures represent the academic tradition of Göttingen and even Germany, and we cannot bow to audience complaints!
We witnessed the birth of a livestreaming legend.

Herman asked, "I have only one question: what if the professor fails?"

Weber shared Lin Ran's view: "Even if the professor fails, he is still a legend."

The progress continued rapidly into the second day, because, to put it simply, the GPY screening method was explained first, and then Zhang Yitang's results were quickly advanced.

On the third night, the number 70000000 appeared on the blackboard.

After fifteen minutes of annotation, the professors in the audience who could understand it were already applauding quietly, with only the action and no sound.

This is already an epic breakthrough for the twin prime conjecture itself.

"Ladies and gentlemen, we have now taken a crucial step toward the ultimate goal of the twin prime conjecture."

We have successfully found a number, a definite number, yes, a number that allows for an infinite number of prime pairs whose difference does not exceed N.

This N is 7000 million.

After Lin Ran finished speaking, the applause from the audience grew louder and louder. Although there were only about a hundred people in the audience, the applause was particularly clear in the middle of the night.

The audience members who were still watching the live broadcast on TV, or those who happened to turn on the TV, were reminded by the applause that something extraordinary had happened.

The mathematics PhDs in the audience heard in the applause that a group of top human brains were cheering for the brilliant insight of another top brain.

After the applause subsided, Lin Ran continued:
"This is the first time in human history that an upper limit has been set for the distance between prime numbers."

Selberg's sieve provides us with a net to capture prime numbers, and the theorem we proved on the first day reveals the secret of the distribution of prime numbers in arithmetic series.

Combining these tools, I have developed a new sieving method in the past two days, optimized the control of the error term, and proved the existence of finite gaps.

This proof is just the beginning! It paves the way for a final solution to the twin prime conjecture, and all that remains is for us to advance N from the current 7000 million to the number explicitly required by the twin prime conjecture—2—in the next three days!

　There's one more chapter today; I'll finish writing this section today.

　　
　
(End of this chapter)