How can a person be struck twice by lightning New Results from Zhang Yitang Yuan Lanfeng

Recently, the most sensational news in the whole mathematical world is that Chinese mathematician Zhang Yitang has once again announced a major result, a breakthrough in the Landau-Siegel problem. Well, the first sentence of this statement is easy to understand, but the second sentence is completely unknown.

 

Zhang Yitang

 

Actually, this news is not a surprise to me at all. Because when I introduced Yitang Zhang in 2019 (There is an upper limit to the minimum interval of prime numbers, there is no upper limit to human struggle | Lanfeng Yuan), I mentioned that he was working on the Landau-Siegel Zeros conjecture and was confident about it. Therefore, the recent news was totally unexpected.

 

But I had not researched exactly what the Landau-Siegel Zeros conjecture was about, and now that he said he had done it, I quickly did some research. However, the conclusion of the research was that it was completely unexplainable to a general audience. Although I've explained a lot of hardcore math problems to people before, this one was really too hard. It's like the classic line from "Yes, Prime Minister": The Russians are too strong.

 

However, I can explain to you clearly the background of this. If you understand this, you are more knowledgeable than 90% of the population.

 

First of all, the basic question is, who is Zhang Yitang? In fact, Yitang Zhang is now very well known as one of the few representatives of the mathematical community who has been through the wringer and is a late bloomer.

 

He was born in Shanghai in 1955, entered Peking University in 1978 for his undergraduate degree in mathematics, and went to Purdue University in 1985 for his Ph.D. After graduating in 1991, he was unable to find a job in academia because his supervisor would not give him a letter of recommendation, so he was reduced to working at various odd jobs to make ends meet. He delivered food, sold fried chicken, and worked as an accountant and cashier in a fast food restaurant. Sometimes he had no place to stay and had to spend the night in his car.

 

In 1999, with the help of his mentor at BYU and professor of mathematics at New Hampshire State University, Liming Ge, he finally got a job in academia at the age of 44: as a temporary lecturer in his mentor's department.

 

Ge Liming

 

In 2005, when Zhang Yitang turned 50, he finally went from temporary lecturer to full lecturer because he taught calculus so well. According to the normal trajectory, it seemed that he would retire smoothly in this position.

 

In 2013, at the age of 58, Yitang Zhang published his stone-cold paper "Bounded gaps between primes", which was the first substantial progress on the twin prime conjecture in centuries.

 

Summary of Bounded Distances Between Prime Numbers

 

What is the twin primes conjecture? This is a problem that can be easily explained with elementary school math knowledge, so let's explain it properly.

 

The top primes are 2, 3, 5, 7, 11, 13, 17, etc., and the intervals between them are 1, 2, 2, 4, 2, 4, etc. Obviously, interval 1 will only occur once, between 2 and 3, because there will be no more even primes after 2. So, how many times will interval 2 occur?

 

An amazing conjecture is: infinitely many times! This is the twin primes conjecture. We call pairs of primes that differ by 2 twin primes, such as 3 and 5, 5 and 7, 11 and 13. The twin primes conjecture says that there are infinitely many pairs of twin primes.

 

In fact, as the natural numbers increase, the primes become sparser and sparser. That is, on average, the separation between two adjacent primes becomes larger and larger. However, the twin prime conjecture says that no matter how large the prime separation becomes, it will always suddenly shrink to 2 somewhere down the line, and this will happen an infinite number of times.

 

Currently, the largest known twin prime pair is 2996863034895 × 2^1290000 ± 1. These are two incredibly large numbers, written in decimal with nearly 400,000 digits.

 

The twin prime conjecture is still a conjecture, which means we don't know if it is right or wrong, and it has neither been proven nor disproven. Yitang Zhang has not proved the twin prime conjecture, but he has proved a weaker version of it. What Yitang Zhang proved is that there are infinitely many pairs of primes whose interval does not exceed 70 million. That is, no matter how large the prime interval becomes, it always narrows down to 70 million or less somewhere behind. If we replace 70 million here with 2, we have the twin prime conjecture.

 

The number 70 million may seem large at first, but it is actually a major breakthrough. This is because it was not known before that there is no upper limit to the interval of primes that occurs infinitely many times, i.e. it is perfectly possible that it is infinite, i.e. the interval of primes becomes larger and larger and larger and no longer shrinks. Zhang Yitang's result is the first time to give a finite upper limit, i.e., to turn infinity into finite. The difference between infinity and 70 million is qualitative, and the difference between 70 million and 2 is only quantitative. Now do you understand why this result is so sensational?

 

Yitang Zhang did not fully exploit the potential of his method. Immediately after the publication of his paper, many mathematicians came to improve it. They organized a collaborative project "PolyMath8" to do this, including the famous Chinese mathematician Zhexuan Tao, a Fields Prize winner.

 

 Zhexuan Tao

 

Through the efforts of so many people, this upper limit has now been reduced from 70 million to 246. i.e., we have proved that there exist infinitely many pairs of prime numbers whose interval does not exceed 246.

 

PolyMath8 project's current record

 

This is not a small improvement, but it is clearly not of comparable importance to the initial proposal of 70 million. Moreover, the potential of this method seems to have been exhausted, not enough to shrink from 246 to 2. If one wants to finally prove the twin prime conjecture, new ideas and new methods should be needed.

 

After the breakthrough of the twin prime conjecture, Yitang Zhang has become a legend. But he did not lie on the credit, he returned to a big problem that he had often thought deeply about before, the Landau-Siegel zero conjecture. As mentioned earlier, I found this one too complex to explain to a general audience.

 

I can only give a little background here. One of the most famous, difficult, and important unsolved mysteries in all of mathematics is called the Riemann hypothesis, which is the key to determining the distribution of prime numbers. I have done six programs before to explain the Riemann Conjecture (Riemann Conjecture (6) proposed in 1859, mathematicians to prove the Riemann Conjecture to what extent? | Technology Yuan Ren), and you are welcome to check it out.

 

Representing Riemann's conjecture in the complex plane: all nontrivial zeros of the Riemann ζ function are on the critical line with the real part equal to 1/2

 

The Riemann conjecture is already quite complicated, but I think after watching these six episodes, the average person can still understand a good deal of it. The Landau-Siegel zero conjecture is a more complicated problem in the context of the Riemann conjecture, so I will drop it completely here. You just need to go through the following narrative to understand the importance of this problem.

 

Yitang Zhang says, "For number theorists, there are two universes. In the first universe, there is no Landau-Siegel zero point. But in the second universe, this zero point exists. Our confusion is that we don't know which universe we actually live inside."

 

His colleague Stopple, a number theorist, explained that if Yitang Zhang could prove the Landau-Siegel zero conjecture, "it would be like the same person being struck twice by lightning" and "if he never became famous, making this work would put him in the world's attention just as much as the last time ".

 

Zhang Yitang is confident that he can solve the Landau-Siegel zero conjecture, believing that there are no major obstacles and that all that remains are just technical problems. Godfrey Harold Hardy (1877 - 1947), the British mathematician, famously said, "More than any other art or science, mathematics is a young man's game." And another famous quote, "I have never seen any mathematician over half a century old start a major mathematical advance."

 

 Hardy

 

Zhang Yitang was asked what he thought of these views of Hardy, and his response was.

 

"This statement probably doesn't apply to me. I still trust my intuition, and I still have confidence in myself. I still have quite a few visions."

 

For now, Yitang Zhang's paper, "Discrete mean estimates and the Landau-Siegel zero," has just been posted online and has not yet been formally submitted, so there is still the possibility of error and it will have to be peer-reviewed before it is finalized. If it is correct, Yitang Zhang was indeed struck by lightning twice, performing one miracle at age 58 and another at age 67. Only for the average person, this latter miracle is much harder to understand than the former one, so we can't explain it in visual terms.

 

Abstract of Discrete Mean Estimation and the Landau-Siegel Zero

 

Although it is impossible to decipher the details, I can still explain to you the nature of this result: as in the previous one, a major breakthrough has been made on a certain problem, but still without solving the problem itself. The Landau-Siegel zero conjecture says that the Landau-Siegel zero does not exist, from which a direct corollary would be derived that some quantity is greater than the inverse of some quantity i.e. -1th power.

 

Corollary to Landau-Siegel's Zero Conjecture

 

Yitang Zhang cannot prove this, but can prove a weaker version by replacing 1 with 2022. yes, it really is the number 2022, and this bright number is no joke!

 

Theorem 1 of The Discrete Mean Estimate and the Landau-Siegel Zero

 

In his paper, Yitang Zhang also points out that the estimate 2022 can probably be improved, just like the 70 million can be improved for the twin primes conjecture. However, at the same time, it is unlikely that 2022 can be improved to 1 by the current idea, just as the 70 million for the twin primes conjecture cannot be improved to 2. Thus, this paper is a significant progress, but not a complete solution, to the Landau-Siegel zero conjecture. However, Yitang Zhang's result is sufficient to prove many outstanding propositions and turn them into theorems, so this result is very significant indeed.

 

Yitang Zhang's assessment of the methodological potential of this paper

 

As I introduced a while ago, Chen Jingrun's contribution to Goldbach's Conjecture is of this nature (16 number-theoretic puzzles, how many can you read and understand? How many can you solve? | Yuan Lanfeng). The goal is "1 + 1", but people used to do "9 + 9" and even "1 + 3", etc. Chen has achieved "1 + 2 Chen Jingrun has achieved "1 + 2", just one step away but still far away. I think, to understand this degree, your knowledge level is more than 99% of people.

 

Chen Jingrun

 

Finally, I would like to say that even if we cannot read these mathematical details, Zhang Yitang's miracle can still give us great inspiration and show us that people can always exert their subjective initiative and always break the boundaries (2019 Technology Yuanren Annual Gala Speech: No One Can Stop You From Striving | Yuan Lanfeng). There is an upper limit to the minimum interval of prime numbers, while there is no upper limit to human struggle. Just as in the "Romance of the Three Kingdoms", Zhuge Liang gave Zhou Yu a sentence in his sacrificial note (Shanxi University made an important breakthrough in quantum science and technology, the beginning does not hang down its wings, but can eventually fight its wings | Yuan Lanfeng).

 

"If you don't hang your wings at the beginning, you will be able to fight for them at the end!"