Shulou(Shulou.com)11/24 Report--

One summer day in 1900, more than 200 of the most outstanding mathematicians held an international conference of mathematicians in Paris, France. At the meeting, Hilbert, a famous German mathematician, gave an important speech entitled "Mathematical problems".

In his speech, he listed a series of math problems that he thought were the most important. Those problems attracted the interest of many mathematicians and had a far-reaching impact on the development of mathematics.

A hundred years later, in 2000, mathematicians from the Clay Institute of Mathematics held a mathematics conference in Paris, France. At the meeting, the participants also listed some of the most important math problems in their opinion. Their prestige was not comparable to that of Hilbert, but they did what Hilbert could not: set up a huge bonus of $1 million for each problem.

In addition to being held in Paris, France, the two echoing mathematical conferences have one remarkable thing in common, that is, one-and only one-of the listed problems is common.

This problem is the Riemann conjecture, which is regarded by many mathematicians as the most important mathematical conjecture.

The Riemann conjecture was proposed by a mathematician named Bernhard Riemann (Bernhard Riemann). Riemann was a German mathematician who died young. He was born in 1826 and died in 1866 at the age of less than 40. Although Riemann's life is short, he has made great contributions to many fields of mathematics, and his influence has even spread to physics. Riemannian Geometry, named after him, is not only an important branch of mathematics, but also an indispensable mathematical tool for Albert Albert Einstein to create general relativity.

In 1859, the 32-year-old Riemann was selected as a member of the Berlin Academy of Sciences. In return for this noble honor, he submitted a paper entitled "on the number of primes less than a given number" to the Berlin Academy of Sciences. The paper, which is only eight pages long, is the birthplace of Riemann's conjecture.

Why is Riemann conjecture the most important mathematical conjecture? Is it because it's very difficult? No.

Of course, the Riemann conjecture is indeed very difficult, it has a history of more than a century and a half since it came out. During this period, many well-known mathematicians made painstaking efforts to solve it.

However, if only measured by the difficulty, then some other famous mathematical conjectures are not inferior. For example, Fermat conjecture was proved after more than three and a half centuries of efforts, while Goldbach conjecture came out more than a century earlier than Riemann conjecture, but it has stood as long as Riemann conjecture. These records undoubtedly represent difficulties, and they may not be broken by Riemann's conjecture.

So what is the reason why Riemann conjecture is called the most important mathematical conjecture? The first reason is that it is inextricably linked with other mathematical propositions.

According to statistics, in today's mathematical literature, there are more than 1000 mathematical propositions based on the establishment of Riemann conjecture (or its extended form). This shows that once the Riemann conjecture and its generalized form are proved, the influence on mathematics will be very great, and all the more than a thousand mathematical propositions can be promoted to theorems; on the contrary, if the Riemann conjecture is overturned, some of the more than a thousand mathematical propositions will inevitably be buried with them. A mathematical conjecture is closely related to so many mathematical propositions, which is unique in mathematics.

Secondly, Riemann conjecture is closely related to the distribution of primes in number theory. Number theory is a very important traditional branch of mathematics, which is called "the queen of mathematics" by the German mathematician Gauss. The distribution of prime numbers is a very important traditional topic in number theory, which has always attracted many mathematicians. This "noble pedigree", which is deeply rooted in tradition, also increases the status and importance of Riemann's guess in the hearts of mathematicians.

Moreover, there is another measure of the importance of a mathematical conjecture, that is, whether it can produce some results that contribute to other aspects of mathematics in the process of studying the conjecture. By this measure, the Riemann conjecture is also extremely important.

In fact, one of the early achievements made by mathematicians in the study of Riemann conjecture directly led to the proof of an important proposition about prime distribution-prime number theorem. Before the prime number theorem is proved, it is also an important conjecture with a history of more than 100 years.

Finally, and most surprisingly, the importance of Riemann's conjecture went beyond pure mathematics and "invaded" the realm of physics. In the early 1970s, it was found that some studies related to the Riemann conjecture were significantly related to some very complex physical phenomena. The reason for this connection remains a mystery to this day. But its existence itself undoubtedly further increases the importance of Riemann conjecture.

For many reasons, Riemann conjecture is well deserved to be called the most important mathematical conjecture.

Image source: the content of pexels Riemann conjecture cannot be described by completely elementary mathematics. Roughly speaking, it is a conjecture for a complex variable function called Riemann Zeta function (a function in which both variables and function values can be valued in the complex field). Like many other functions, the value of Riemann Zeta function is zero at some points, and those points are called zeros of Riemann Zeta function. Among those zeros, there are some particularly important non-trivial zeros called Riemann Zeta functions.

What Riemann conjectures is that those non-trivial zeros are all distributed on a special straight line called the "borderline". Riemann's conjecture is still up in the air (neither proved nor overturned) to this day.

However, mathematicians have studied it deeply from two different aspects: analysis and numerical calculation. The strongest result in analysis is to prove that at least 41.28% of the non-trivial zeros are on the critical line, while the strongest result in numerical calculation is to verify that the first 10 trillion non-trivial zeros are all on the critical line.

Wen Yuan: the book "Pauli's mistake: flowers and Grass in the Hall of Science" author: Lu Changhai part of the picture source network copyright belongs to the original author Editor: Zhang Runxin this article comes from Wechat official account: Origin Reading (ID:tupydread), author: Lu Changhai, Editor: Zhang Runxin

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