Chapter 196: Day3
After Lin Ran finished speaking, the number theory experts present collectively stood up and applauded.
Other PhD students or mathematics professors from other subfields didn’t quite understand, but they also politely applauded.
For a moment, the reporters present were baffled.
After their applause subsided, they found a professor who had stood up nearby and asked in a low voice:
“I want to ask, is this very impressive?”
“Extremely impressive. The professor has found another cornerstone for analytic number theory. These two results he presented today can not only be used for the twin prime conjecture but also for other number theory problems.
The professor not only found the modulus but also expanded the modulus range. The increase in the modulus range directly enhances the ability of the sieve method and distribution analysis.
The bilinear form and divergence technique he used in it provide us with new tools for analytic number theory and the sieve method.
In short, this is already an incredibly awesome achievement.
A general mathematician could win the Fields with this. Just having this result makes the trip to Göttingen worthwhile.”
Enrico Bombieri won the 1974 Fields precisely with this achievement. He extended the modulus range from 1/2 to 4/7, and major improvements to the standard theorem didn’t come until 1987.
Lin Ran’s current content is equivalent to at least two Fields.
However, this is merely the beginning.
The mathematicians present, as long as they work in number theory, felt like they were about to climax.
“Good, everyone. We’ve now advanced the modulus to four-sevenths. Sorry, time is tight, so I won’t discuss it.
If you have questions, note them down first. I’ll try to answer. If there’s no time this time, I’ll do Q&A again at Columbia University when I return to New York.”
Fox shouted loudly from the audience: “Good, no problem, professor. Please continue.”
Döblin was speechless. Why are you shouting? This isn’t your home turf. Are all Americans so annoying!
However, considering this was an unprecedented occasion and time, he didn’t lose his temper.
“We’re going to continue advancing now.”
Lin Ran wrote a new formula on the blackboard:
This formula, sixty years later, is called the Elliott-Halberstam conjecture. The EH conjecture was proposed by Elliott and Halberstam in 1968 and published in Symposia Mathematica. It remained unproven until 2025.
To put it simply, proving this conjecture means the distribution error of primes in arithmetic progressions with modulus ≤1 can be effectively controlled, far surpassing the standard theorem’s one-half.
The K=246 for twin primes could quickly be advanced to K=6, almost just one step away from the K=2 needed for the twin prime conjecture.
Like Nathalie Debouzy’s 2019 achievement, which, by improving the asymptotic sieve method and assuming the EH conjecture, shows there are infinitely many almost twin primes. What are almost twin primes? It means p is prime, and p-2 is prime or semiprime.
The EH conjecture is so important that later mathematicians have even started assuming it holds.
In other words, Lin Ran can no longer rely on later wisdom and must completely prove the EH conjecture on his own.
It can even be said that the EH conjecture is the one where the modulus approaches 1 infinitely, and pushing the EH conjecture further, i.e., directly to 1, requires an entirely new mathematical framework.
Therefore, after entering this stage, Lin Ran’s pace clearly slowed down.
Because even more critically, Lin Ran couldn’t directly use theorems or lemmas available sixty years later; all the tools needed sixty years from now had to be reinvented on the spot in the auditorium of the University of Göttingen.
“Bilinear form and divergence technique—no, this can at most advance to four-sevenths.”
“Type II estimate, relying on short interval distribution control and smoothed modulus optimization—no, it still can’t reach this level.”
“L-function zero relationships could be a path.
The EH conjecture involves the error term for average modulus q, and each q corresponds to a Dirichlet character χ(mod q), whose L-function zeros affect the distribution.
The proof of the Bombieri-Vinogradov theorem relies on zero density estimates, controlling the number of L-function zeros near Re(s)≈1.
The EH conjecture requires stronger zero control, which involves the distribution laws of zeros in the critical strip. Then leveraging indirect support from GRH.”
Lin Ran wrote on the blackboard and erased, erased and wrote again.
The scholars present all knew very well that this problem was very important.
Just this conjecture itself was already extremely valuable.
Continuing until eleven at night, Lin Ran began to speed up the rhythm of writing with chalk, without a single pause.
The student next to him responsible for changing the blackboards had already changed twice.
He wrote nonstop, filling a full thirty blackboards.
There were only about twenty professors sitting in the audience, and even more people lying on the ground in sleeping bags.
As the sound of chalk friction on the blackboard became more pronounced and faster, those awake woke up those who were sleeping.
Everyone paid attention to the content on the blackboard.
“This is?”
“Exactly, Randolph has found the way out.”
“We are indeed witnessing history. The twin prime conjecture is just the final destination; we’re now appreciating the scenery along the way to the destination.”
“I just fell asleep. Which path did Randolph choose?”
“I think it’s extending the differential distribution describing zeta function zeros to Dirichlet L-functions to influence the average behavior of arithmetic progressions. If the zero distribution fits the random matrix model, then it means it can support the error control for his conjecture.”
“This is an approach, but whether it’s feasible still depends on his specific design.”
Lin Ran finished writing and looked at the achievement in front of him, feeling a genuine sense of accomplishment:
“Alright, that’s it for today.
Everyone can take a look; I’m already extremely sleepy.
The current result deepens our understanding of prime distribution and creates prerequisite tools for the proof of the Twin Prime Conjecture.
Its breakthrough lies in surpassing the limitations of past moduli.
For the proof process of this conjecture at the end, I analyzed the nontrivial zero distribution of the Dirichlet L-function.
By assuming the zeros are sufficiently sparse within the critical strip, I estimated the average behavior of the error term. Then I designed a new sieve method, combining bilinear form estimates and divergence technique, to optimize modulus decomposition and break through the bottleneck of traditional methods.
Finally, through a new lemma, I controlled the high-dimensional exponential sums to ensure the error term satisfies the conjecture’s requirements.”
Lin Ran finally added some annotations on the blackboard.
“Everyone, I’m going to sleep first; we’ll continue in about six hours.”
Lin Ran didn’t leave; he went directly to the small room next to the auditorium to rest.
The professors and PhDs in the audience had all crowded to the front to look at the content on the blackboard.
Over the whole day today, Lin Ran had written a full thirty blackboards.
The Bombieri-Vinogradov theorem and the strengthened form of the Bombieri-Vinogradov theorem were easy to understand.
Moreover, Princeton had already produced the Bombieri-Vinogradov theorem itself, so they understood the Bombieri-Vinogradov theorem and its strengthened form very quickly.
It reached the EH conjecture.
Because the EH conjecture itself didn’t exist yet at this point, Lin Ran had essentially handled everything from proposing the conjecture to proving it himself.
“Too beautiful, simply a work of art.”
“This is a super-enhanced achievement.”
“Is there room for simplification here?”
“No, zero density estimate and pair correlation conjecture might simplify the professor’s proof of this conjecture, but we still need to think carefully.”
“The angle of controlling high-dimensional exponential sums to ensure the error term satisfies the conjecture’s requirements is too ingenious.”
“No, I need to hurry back and send today’s achievement to colleagues still at the school.”
It’s not very realistic to send mathematics papers by telegraph.
In theory, mathematics papers can be simplified to plain text, encoded into ASCII or Baudot characters, and sent in segments via teletype, but in practice, it’s extremely difficult to express precisely.
Now, people generally scan papers with fax machines and send the diagrams directly.
Due to the high cost of faxing, even in Göttingen as a university town, there were only a few fax machines.
But Lin Ran’s result today was so astonishing that whether it was the result itself or the methods used, it thrilled number theory scholars, who wanted to share it with colleagues at their own school right away and urge them to come to Göttingen to witness the miracle.
Vacation? At a time like this, what vacation? Coming to Göttingen to witness the miracle on site is most important.
Even though it was late at night, the scholars present, whether they had rested earlier or not, were now full of energy.
Considering that Lin Ran had already gone to the rest room next door to rest, they discussed today’s achievement in lowered voices.
“No matter what, just advancing the modulus to this extent in the earlier part is already a tremendous achievement.”
“Tremendous? At least it’s one of the most important achievements in the number theory field this century.” Döblin corrected.
“Professor Seagull, no no no, it’s not time for final conclusions yet; there are still five days left. From today’s achievement, Randolph might still complete the Twin Prime Conjecture.
In some sense, the Twin Prime Conjecture is comparable to the Goldbach Conjecture.
If he can complete it, then this is undoubtedly the most important achievement in number theory this century; we can drop the ‘one of’.”
“Unless in the remaining thirty-five years of this century, someone completes the proof of the Goldbach Conjecture.” said Jean Pierre, the top mathematician from France (won the Fields at 27, collecting the three major mathematics awards).
He had previously worked on commutative algebra and algebraic topology, but in the past decade or so, he had turned to collaborating with Grothendieck on algebraic geometry.
He continued: “I used to think number theory was too simple; there are too many tools from analysis that can be applied in the number theory field.
I think we number theorists now are far behind Randolph in mastery of analysis.”
Seagull smiled wryly: “We can’t use Randolph as the standard to demand of young students; that would be too much.”
Pierre shook his head: “No, I’m not saying to demand Randolph’s solid foundation in analysis from young PhDs, but in training young scholars, we can’t relax the requirements on their analysis ability just because their topic is number theory.
Number theorists should have analysis ability comparable to PhDs in analysis; PhDs in analysis should have understanding of algebra comparable to PhDs in algebra. Ever since Randolph proposed the Randolph Program, we’ve increasingly realized the very close connections between different subfields of mathematics.
We should push young scholars not to relax in any area; we need to cultivate all-round scholars.”
Seagull said: “You plan to go back to Paris Normal School and push for requiring students in Paris universities’ mathematics departments to meet this standard?
Isn’t that going backwards again?”
What Seagull meant by “going backwards” here was that early on, everyone trained all-round mathematics PhDs, but later, as modern mathematics became more abstract and people’s energy was limited, they no longer required students to master everything, since ordinary people couldn’t do it.
“Exactly, let’s try it; at least we should demand this standard from the most talented batch of young people.” Pierre nodded, his gaze still fixed on the densely packed formulas on the blackboard.
Seagull was older than Pierre and fully understood what the other wanted to express: “You mean mathematics is changing, and our concepts for training talent need to change too?”
Pierre nodded: “That’s right, Randolph revealed the connections between different branches of mathematics through his actions and achievements, and these connections are becoming increasingly evident. During my communication with Grothendieck, we both thought the same.”
Maybe it’s Randolph, maybe it’s someone else, but someone will turn the concepts proposed in the Randolph Program into reality.
Standing at the current moment when modern mathematics is about to enter a new stage, we should also promptly change the strategy for training young mathematicians.
Additionally, my selfish motive is that I also want to unearth talent like Randolph for France.”
“No audience wants to watch such a television program. The professor barely speaks, and when he does, it’s highly professional mathematical knowledge. Don’t even mention me; even the PhDs in Mathematics and Mathematics Professors on site, few can fully understand what the professor is saying.
Our audience wants more information. Our current live broadcast is completely wrong.” Program Director Hermann Schmidt said.
That night late, besides the mathematicians who couldn’t sleep, the NDR staff were also sleepless.
Except for the large audience at the start of the live broadcast, the numbers dwindled afterward.
Because everyone couldn’t understand it.
The NDR Hanover branch meeting room was filled with smoke, with documents, ashtrays, and coffee cups scattered on the wooden table, and the NDR logo and program schedule hanging on the wall.
Hermann continued: “We have to solve this problem.
The professor’s lecture has only aired for one day, and viewer complaints are piling up on the table.
Some say it’s alien language, others threaten to change channels.
We can’t just sit by; what about the six days of live broadcast?”
Hermann’s hair was nearly falling out.
They thought it would be an unprecedented explosive live broadcast, and it was explosive, but only for the first half hour.
After half an hour, it was all complaints.
“The telephone hasn’t stopped since the broadcast started. Viewers say they want to see the professor speak, but not this kind of lecture full of formulas on screen.
Someone even asked if we aired the wrong channel!
We need to take action, or the audience will be lost.
Can we communicate with the professor to have him talk about fun stories from the moon landing? Talk about aerospace and the moon. Now the on-site challenge is the Twin Prime Conjecture. I admit it’s very significant, with global mathematicians following, but have we overestimated the audience’s acceptance?”
Station Manager Webber had arrived at the office half an hour ago. He said helplessly: “Who will ask the professor to adjust the content?
Will you go? Do you have the face to do it?
It’s clear the professor is now set on creating a miracle in Göttingen. The person who can change his mind is in the White House. You need to find President Lyndon Johnson first, then have him negotiate with the professor.
None of us can do that.
Moreover, this is a historic event. NDR has the responsibility to broadcast such an academic spectacle. We can’t abandon it because of viewer opinions.
This way, we can pre-record some commentary segments to air before and after the daily live broadcasts, explaining basic concepts. After the professor finishes, contact the Berlin side and have them send Mathematics Department professors for support.
We need someone more professional than us but able to express in popularized language better than the professor himself to explain to the audience.
In short, Hermann, remember, NDR is not an entertainment channel. We are public broadcasting, and education is the core mission. Randolph’s lecture represents the academic tradition of Göttingen and even Germany. We can’t bow to viewer complaints!
We are broadcasting the birth of a legend.”
Hermann asked: “I have only one question: what if the professor fails?”
Webber’s view was the same as Lin Ran’s: “Even if the professor fails, it’s still a legend.”
After entering the second day, progress was still rapid, because basically, they first explained the GPY sieve method once, then quickly advanced to Zhang Yitang’s achievement.
On the evening of the third day, the number 70000000 was already on the blackboard.
After fifteen minutes of annotation, the professors in the audience who could understand were already applauding quietly, with movement but no sound.
For the Twin Prime Conjecture itself, this was already an epic breakthrough.
“Everyone, we have now taken a key step toward the ultimate goal of the Twin Prime Conjecture.
We have successfully found a number, a specific number. Yes, this number ensures there are infinitely many prime pairs whose difference does not exceed N.
This N is 70 million.”
After Lin Ran finished speaking, the applause in the audience grew from small to large. Though there were only about a hundred people, the applause was exceptionally clear in the late night.
At this moment, the viewers still persisting in front of the television, or those who happened to turn it on, were reminded by this applause that something extraordinary was happening.
The PhDs in Mathematics on site heard from this applause a group of humanity’s top brains cheering for the flash of wisdom from another top brain.
After the applause subsided, Lin Ran continued:
“This is the first time in human history that an upper bound has been set for the distance between primes.
Selberg’s sieve method provided us with a net to capture primes. The theorem we proved on the first day revealed the mystery of prime distribution in arithmetic progressions.
Combining these tools, I developed a new sieve method in the past two days, optimizing the control of the error term and proving the existence of finite gaps.
This proof is just the beginning! It paves the way for the ultimate solution to the Twin Prime Conjecture. What remains is to advance N from the current 70 million to the number 2 explicitly required by the Twin Prime Conjecture in the next three days!”
There is still one chapter today; this segment will be finished today.
