Technology invades the modern world

Chapter 221 Fields: The Predictions Are Over
"No, Randolph, I know you're very confident."

When I was about your age, I solved a major problem in mathematical physics, achieving a breakthrough, and I was just as confident as you were.

I felt like the Nobel Prize in Physics was beckoning me. Fields was nothing special, but now I'm over sixty. Without Fields, there would be no Nobel.

It's good for young people to be confident, but the gravitational pull of reality will pull you back to the ground.

Also, Randolph, I'd like to remind you that the next International Congress of Mathematicians is in 2022, two years from now.

Six months later, no Fields Medal will be awarded at that time.

Mikhail Lyubich had to admit he was starting to like this guy a little.

The other person was completely different from any other Chinese or Chinese-American he knew.

New York City has the largest Chinese population in the United States, and there are simply too many Chinese and Chinese international students at Stony Brook University.

Over the years, Mikhail Lyubich has encountered many Chinese students, colleagues, and peers at academic conferences.

The other Chinese people were very humble.

Even Chinese mathematicians who can give an academic presentation for a maximum of one hour at the main venue of the International Congress of Mathematicians will say, when faced with praise from their peers, that they still have a lot to improve on and that many of their peers are more capable than them.

This is the first time I've seen someone as arrogant as Lin Ran.

"Professor Lyubich, what I mean is six months. Just six months is enough to make the 2022 Fields Medal a foregone conclusion," Lin Ran said calmly.

Mikhail Lyubich didn't dwell on it any longer. He didn't want to discourage the young man when he was at his most arrogant. He had already given him all the necessary reminders: "I'm looking forward to that moment."

Lin Ran said, "Professor Lyubich, you'll see soon enough, but I still need the school to provide me with a little help."

Mikhail Lyubich said: "Of course, this is the school's obligation."

Tell me, which professor would you like to mentor you? To be honest, I've been thinking about this for the past few days. You know, Stony Brook isn't very good at number theory.

The intersection of algebraic geometry and number theory is even less.

Only a handful of professors may have done some related work, and the relevance is very weak, making it difficult to provide you with effective guidance.

Before Mikhail Lyubich could finish speaking, he wanted to say that the good news was that Lin Ran wouldn't have to wait too long, as they had recruited a professor from England whose research topic overlapped with Lin Ran's and who would be on duty soon.

Lin Ran said, "No need, that's not what I meant."

What I mean is that my influence is currently limited, and I need to give academic presentations on some of my cutting-edge mathematical work. I can't keep bothering Terry, so I hope Stony Brook can help me organize enough influential mathematicians to attend my presentations.

This way, I can spread my reputation throughout the mathematics community as quickly as possible.

The professor's brilliance will shine upon the entire mathematics community in 2020.

Mikhail Lyubich asked directly: "So Randolph, hasn't your weak form proof of Goldbach's conjecture already undergone peer review?"
"Have you achieved something new again?"

Mikhail Lyubich could roughly guess where Lin Ran's confidence came from; it must be from results.

Only those who have achieved results can be so confident.

In other words, Lin Ran still has some results to show for it.

Lin Ran said, "Of course, I will release a series of results in the next six months."

I hope the Stony Brook campus can help me organize a Zoom meeting.

My idea is to first use pure mathematical tools to extend the unconditional bound of the twin prime conjecture to 246.
Then we can completely eliminate the twin prime conjecture using pure mathematical tools.

In this process, my mathematical ideas and concepts need to be disseminated in the mathematics community through numerous mathematics conferences.

In this regard, we need to rely on the Stony Brook University campus.

Mikhail Lyubich was shocked.

That was truly shocking.

Because Zhang Yitang proved the limit of the twin prime conjecture in 2013, and then in 2014, mathematicians around the world used computer-aided tools to reduce this limit to 246.
Without computer assistance, the maximum number that can be achieved is 600, a fact proven in November 2013.

In other words, from November 2013 to today, a total of more than six years, many mathematicians have tried to prove the upper bound of the twin prime conjecture 246 using pure mathematical tools.

But more than six years have passed, and no one has succeeded.

(P.S.: Not to mention more than six years, no one has achieved this even by 2025.)

Mikhail Lyubich then thought that the weak form of Goldbach's conjecture also utilizes a cross-disciplinary approach: "So the method you used in Goldbach's conjecture can also be used in the twin prime conjecture?"

"No, no, no, that method can't be used here."

Unlike Goldbach's conjecture, the twin prime conjecture uses purely mathematical tools to prove the upper bound of the twin prime conjecture 246, which is a purely number theory technique.

It only needs to use the core ideas of number theory; it doesn't need to be that complicated.

However, completely solving the twin prime conjecture requires some interdisciplinary knowledge.

Mikhail Lyubich asked skeptically, "When do you plan to start?"

Lin Ran said, "We can start anytime, depending on when the Stony Brook branch can arrange things for me."

"Arrange the time. I'll upload the paper to ArXiv three days in advance, and then we can start the presentation three days later."

Mikhail Lyubich said, "Then post it today, three days from now. Don't worry about the Zoom meeting; even with the time difference, I'll arrange everything for you."

Whether it's a mule or a horse, you have to put it to the test.

If you can prove yourself, then Stony Brook University will receive more resources.

Clearly, pure mathematical tools have lowered the limit of the twin prime conjecture to 246, which is sufficient for publication in the top four journals.

There are quite a few PhDs who can publish in the Big Four accounting firms globally, but when that number becomes two, very few can do it.

To publish two articles in the top four journals within a month is even rarer.

Everyone rushes to publish their findings as soon as they have them; how could anyone keep their results to themselves?

A PhD who can publish in the Big Four accounting firms has no trouble finding a job.

There's no need to worry about the "publish or perish" mentality or holding onto research findings until universities publish them.

"Lin Ran, wait, why the change of heart again? Weren't you going to replicate the Apollo project? How come you've suddenly become a great mathematician?"

Li Xiaoman returned to New York from the Buffalo campus, weary from her journey.

As mentioned earlier, although both are State University of New York, the campuses at Buffalo and Stony Brook are quite far apart.

Because of the virus outbreak and the need for remote learning, Li Xiaoman originally wanted to come back, but she hadn't made up her mind before.

After seeing Lin Ran's WeChat message, she resolutely returned home, even though she had to face her annoying aunt and uncle.

"How come I didn't know you had such talent!" Li Xiaoman was truly shocked.

Even after seeing Lin Ran's WeChat message, she still knew about Fields.

Then she searched online.

You wouldn't know unless you searched.

I was shocked when I searched it up.

In just over a month since they parted ways, Randolph Lin has managed to make such a big splash.

He's made a name for himself in the mathematics world, hailed as a rising star among Chinese mathematicians. Wait, didn't we initially agree to fool Bezos? How come you've suddenly taken off in academia?

Even though Li Xiaoman is a liberal arts student, she still has a basic understanding of mathematics.

Knowing what Lin Ran's achievements mean, at least the path to the top in the mathematics world has been paved.

Lin Ran took a deep breath and then said, "In short, a small accident happened."

However, the moon landing still had to be done, although there was a slight setback along the way.

Believe it or not, I've always been very interested in math.

Li Xiaoman interrupted, "Stop! You said before that you were very interested in the Apollo moon landing."

Now it's back to math, isn't it?

Lin Ran shook his head and said, "I am also interested in mathematics."

In the process of thinking about the Apollo program recently, I have come to understand many mathematical problems that I couldn't understand before.

My brain works better now, can you understand?

Previously, my level in mathematics was only good enough to be an enthusiast; now, I'm more than qualified to be a mathematician.

Lin Ran was very reserved in front of Li Xiaoman. He was a mathematician, but he didn't call himself a master.

Li Xiaoman sighed: "So, will Bezos still not see us?"

However, don't get your hopes up too high, because given the current environment, the fundraising dinner will probably have to be moved online.

Democratic senators certainly wouldn't dare to risk public outrage by hosting an in-person dinner.

This also means you might not get the chance to speak with Bezos one-on-one.

Lin Ran nodded and said, "Sister Xiaoman, if there's a chance, then let's meet; if not, then forget it."

So what exactly are your plans now?

Lin Ran said casually, "Let's get Fields first."

(Citation chart of all papers related to the twin prime conjecture)

"A mathematical proof of the twin prime conjecture and the fact that the distance between prime numbers does not exceed 246"

That evening, Randolph Lin's Arxiv homepage featured a new article.

The article title is nothing new.

However, the article's abstract is quite original.

Because the abstract clearly states:
"Past proofs of the twin prime conjecture relied on key mathematical tools such as the Selberg sieve, the GPY sieve, and the multidimensional Selberg sieve. These works optimized its limits, but the numerical optimization schemes used could not be separated from computationally intensive work. Computationally intensive optimization methods allow the parameters of k-tuples to be handled by computers."

The author believes that by utilizing the EH conjecture, leveraging stronger zero-point control, and further aided by the indirect support of GRH, the problem can be effectively solved.

The entire abstract can be summarized in one sentence.

We can use purely mathematical methods to reduce the limit of the twin prime conjecture to 246.
Terence Tao was very excited after reading it.

He was the one who proposed the collaborative research on the twin prime conjecture back then, and he has a special affection for this problem.

In addition, Randolph was one of the new talents in the mathematics world that he discovered.

He called James Maynard, his partner in the twin prime conjecture: “James, have you read Randolph’s new paper? It’s amazing.”

Tao was thrilled by this breakthrough, recalling the efforts made on the Polymath project six years ago.

James Maynard's exclamation came from the other end of the phone: "Of course, Randolph's hot right now."

As soon as his new paper was posted, a colleague messaged me on WhatsApp, urging me to take a look.

I just read the abstract. He claims to have proven, using purely mathematical tools, the existence of infinitely many pairs of prime numbers with a gap of no more than 246. This result is truly astonishing.

I could only manage 600 before, but he managed to get it to 246.
However, I didn't see exactly how he did it.

I still have some doubts about this result.

Because he had done this kind of problem before, he understood exactly how difficult it was.

I can only manage 600, but you managed 246.

Terence Tao explained, "Based on our work on the Polymath project, he introduced some new technologies to further optimize the sieve weights."

From my cursory reading of the paper, he seems to have handled the error term more effectively, possibly using some advanced analytic number theory tools, such as improved Fourier analysis.

However, I still have many questions. I'm very curious about how Randolph specifically did it.

A complex summation estimate came to Tao's mind; he had been trying to deduce how Randolph optimized the error term.

James Maynard exclaimed excitedly, "That sounds interesting! I'll have to find time to read his paper carefully."

However, their confusion did not last long.

Three days later, Stony Brook University held a Zoom meeting where Lin Ran explained his latest findings to the mathematicians.

For the next six months...

Lin Ran's progress can be described as a lightning-fast, rapid advance.

A month later, the proof of the EH conjecture was released.

As mentioned earlier, the EH conjecture was proposed by Elliott and Halberstam in 1968 and published in Symposia Mathematica. Originally, the conjecture had not been proven until 2025.

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.

As a hypothesis that had been dormant for more than fifty years, the paper caused a sensation as soon as it was released.

Randolph was no longer a newcomer to the mathematical community, thanks to his two papers improving upon the work of others.

With two papers published in the Big Four accounting firms, it's no exaggeration to describe him as a rising star in the mathematics world.

After his paper was published, many people thought that if the EH conjecture was proven, the twin prime conjecture would also be solved soon.

To a certain extent, the two are equivalent.

However, they didn't have time to react.

Lin Ran iterated on this paper about the proof of the EH conjecture.

A Proof of the Twin Prime Conjecture Based on the Proof of the EH Conjecture

Public opinion was in an uproar for a while.

In the first half of 2020, in addition to the virus raging around the world, Randolph also wreaked havoc in the mathematics community.

The first two are results from four major journals, and the latter two are Fields Medal winners.

It was published so casually after only six months.

After the proof of the twin prime conjecture was published, every media outlet that covered academic news went crazy asking the experts for their opinions.

The most frequent question is whether the 2022 Fields Medal has already lost its suspense.

(End of this chapter)