This article is about 1800 words, and there are a lot of formulas. Please read it just by yourself.
Recently, in the scientific community has doubled and made a super big news. Professor Zhang Yitang , Department of Mathematics, University of California, Santa Barbara, USA, announced that he has made important progress in the mathematical problem of Landau Siegel zero-point conjecture.
Zhang Yitang
Although most of us may not know who Landau ( Note: This Landau is not the great Soviet physicist Landau, but the German mathematician Landau ) and Siegel are. However, with such huge academic news on the Internet recently, I wonder if you are interested in preparing a bench for melon seeds?
Riemann ζ function
First of all, we need to introduce a Riemann ζ (ζ is pronounced as zeta) function . At the beginning, this function is like this
For example, add up infinite fractions can actually be related to pi. Isn't it quite magical?
But when s=1,
Isn't this the famous harmonic series? I believe you have learned all the advanced mathematics of in the freshman year. Obviously, this series is divergent, that is, this series is equal to infinity, and its proof is also very simple.
In this way, as long as s≤1, the Riemann ζ functions will all be infinite. How can I play this?
In order to solve this infinity problem, Riemann made a clever parse extension , successfully extending the definition domain s of Riemann ζ function to all plural numbers,
For example,
, and according to our original function,
So many people often say that
So, +2+3+4+‧‧=-1/12 is wrong, because the initial function cannot be smaller than 1, so how can it be -1?
Landol-Segel Zero Point Conjecture
For the Riemann ζ function above, Riemann proposed Riemann conjecture : All non-trivial zeros of the Riemann ζ function are on a straight line with the real part equal to 1/2.
The so-called "non-trivial" can be simply understood as the non-real solution of the Riemann ζ function.
Once the Riemann conjecture is correct, we can accurately calculate the prime number distribution, and people have established many theorems based on the correct Riemann conjecture.
However, people are not satisfied. In order to calculate the prime number distribution in the arithmetic sequence, people further promoted the Riemann ζ function .
At this time, the molecule 1 of the Riemann ζ function is replaced with a more general function, and the Riemann ζ function becomes the Dirichlet L function, and
This function can also be parsed into functions that extend into the entire complex plane. For Dirichlet L function, there is also an generalized Riemann conjecture : all non-trivial zeros of Dirichlet L function are on a straight line with the real part equal to 1/2. Riemann Conjecture is just a special case of the Riemann Conjecture in general.
Landol-Sigel zero point conjecture means: Dilicre L function may have some zero points not on the straight line with the real part equal to 1/2, but are near 1.
Obviously, if the Landau Siegel zero-point conjecture is correct, then the generalized Riemann conjecture will be overturned, which will threaten the correctness of Riemann conjecture ( Personally believe that it is different to overturn the Riemann conjecture. After all, commonality does not represent personality. Those who study mathematics can popularize ), and step by step threaten the correctness of many mathematical theorems.
So, in the heart of mathematicians, it is best not to exist on the Zero Point of Landau Siegel. After all, mathematicians don’t want their previous work to be overturned, right?
What did Zhang Yitang's paper write
In fact, Zhang Yitang's paper is indeed aimed at proving that Landau Siegel does not exist at zero.
At present, Zhang Yitang's paper on the Landau Siegel Zero Point Conjecture can be downloaded on the Internet, with a total of 111 pages. And unlike our usual academic papers, these 111 pages do not have any images or tables, all of which are formula derivation, which is very useful. Interested friends can take it over and take a look.
Zhang Yitang's paper homepage
Zhang Yitang summarized his main conclusions into two theorems.
The final requirement to prove that the Landau Siegel zero point does not exist is to prove that
And Zhang Yitang's paper only proves that
At the same time, the paper pointed out that although the index −2022 in the formula can be replaced with a larger negative value, according to the current method of the paper, the index should not reach the final -1.
Gents with good English will definitely understand
In other words, this paper did not fully prove that the Zero of Landau Siegel did not exist, but was just a big step in proving that the Zero of Landau Siegel did not exist.
If you want to achieve the final -1, future generations need to continue to work hard!
Summary
Maybe many people come in with a melon-eating mentality, and when faced with the formula of the article, they can only slide to the bottom quickly. In this regard, I directly sorted out the key points of this article:
. The content of the Landau Siegel zero point conjecture is that there may be some zero points in the Dirichlet L function that are not on the straight line with the real part equal to 1/2, but are near 1.
. Zhang Yitang has not completely solved the Landau Siegel zero-point conjecture, so everyone does not have to be disappointed, and there is still a chance to solve it yourself.
. Zhang Yitang's paper is 111 pages long, all of which are formula derivation. The correctness of the paper still needs to be tested by the academic community. Friends with strength can try to be reviewers, and maybe they can find problems, right?
I hope the above scattered thoughts will allow everyone to successfully show their math boss' temperament next time they talk about the Riemann conjecture and the Landau Siegel zero-point conjecture.
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does not represent the position of the Institute of Physics, Chinese Academy of Sciences
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Source: Yuzhi
Editor: Miss Zhou πhtml
