One summer day in 1900, more than two hundred of the most outstanding mathematicians held an International Congress of Mathematicians in Paris, France. At the meeting, the famous German mathematician Hilbert gave an important speech entitled Mathematical Problems.
In his speech, he listed a series of what he considered to be the most important mathematical problems. Those problems attracted the interest of many mathematicians and had a profound impact on the development of mathematics.
One hundred years later, in 2000, mathematicians from the Clay Institute of Mathematics in the United States also held a mathematics conference in Paris, France. At the meeting, the participants also listed some of the most important mathematical problems in their opinion. They couldn't match Hilbert's prestige, but they did one thing Hilbert couldn't: set up a massive $1,000,000 prize for each problem.
In addition to being held in Paris, France, these two parallel mathematics conferences have one thing in common, that is, among the listed problems, there is one - and only one - in common. This conundrum is the Riemann Hypothesis, which is considered by many mathematicians to be the most important mathematical conjecture.
The Riemann Hypothesis was proposed by a mathematician named Bernhard Riemann. Riemann was a German mathematician who died young was born in 1826, and died in 1866, at the age of less than 40. Although Riemann's life was short, he made great contributions to many fields of mathematics, and his influence was so wide that it even affected physics. For example, Riemannian geometry named after him is not only an important branch of mathematics but also an indispensable mathematical tool for Albert Einstein to create the general theory of relativity.
In 1859, at the age of 32, Riemann was elected a corresponding member of the Berlin Academy of Sciences. In return for this high honor, he submitted a paper entitled On the Number of Prime Numbers Less Than a Given Value to the Berlin Academy of Sciences. That short 8-page paper is the "birthplace" of Riemann's conjecture.
Why is the Riemann Hypothesis the most important mathematical conjecture? Is it because it is very difficult? no.
Of course, the Riemann Hypothesis is indeed very difficult. It has a history of more than one and a half centuries since it came out. During this period, many well-known mathematicians put in painstaking efforts to try to solve it.
But other well-known mathematical conjectures aren't far behind when measured solely in terms of difficulty. For example, Fermat's conjecture was proved after more than three and a half centuries of hard work; Goldbach's conjecture came out more than a century earlier than Riemann's conjecture, but it still stands today just like Riemann's conjecture. These records undoubtedly represent difficulties, and the Riemann Hypothesis may not be broken.
So, what is the reason why the Riemann Hypothesis is called the most important mathematical conjecture? The first reason is that it is inextricably linked with other mathematical propositions.
According to statistics, there are more than a thousand mathematical propositions in today's mathematical literature that are based on the establishment of the Riemann Hypothesis (or its extended form). This shows that once the Riemann Hypothesis and its extended form are proven, it will have a huge impact on mathematics, and more than a thousand mathematical propositions can be promoted to theorems; on the contrary, if the Riemann Hypothesis is overthrown, then It is inevitable that some of the more than a thousand mathematical propositions will be buried with him. A mathematical conjecture is closely related to so many mathematical propositions, which can be said to be unique in mathematics.
Second, the Riemann Hypothesis is closely related to the distribution of prime numbers in number theory. Number theory is an extremely important traditional branch of mathematics, and was called the queen of mathematics by 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 deeply rooted in tradition has also increased the status and importance of the Riemann Hypothesis in the minds of mathematicians to a certain extent.
Furthermore, there is another measure of the importance of a mathematical conjecture, that is, whether some results that contribute to other aspects of mathematics can be produced in the process of studying the conjecture. Measured by this standard, the Riemann Hypothesis is also extremely important.
In fact, one of the early results of mathematicians working on the Riemann Hypothesis led directly to the proof of an important proposition about the distribution of prime numbers—the prime number theorem. Before the prime number theorem was proved, it was also an important conjecture with a history of more than 100 years.
Finally, and most unexpectedly, the importance of the Riemann Hypothesis even goes beyond the scope of pure mathematics, and "invades" the territory of physics. In the early 1970s, it was discovered that some research related to the Riemann conjecture was significantly related to some very complex physical phenomena. The reason for this association remains a mystery to this day. But its existence itself undoubtedly further increases the importance of Riemann's conjecture.
For many reasons, the Riemann Hypothesis is well-deservedly called the most important mathematical conjecture.
The content of Riemann's conjecture cannot be described by completely elementary mathematics. Roughly speaking, it is a conjecture for a function of complex variables (a function in which both the variable and the function value can take values ​​in the field of complex numbers) known as the Riemann zeta function. The Riemann zeta function, like many other functions, has a value of zero at certain points, known as the zeros of the Riemann zeta function. Among those zeros are a particularly important subset of nontrivial zeros known as the Riemann zeta function.
What the Riemann Hypothesis guesses is that those non-trivial zeros are all distributed on a special straight line called the "critical line". The Riemann Hypothesis remains unsolved (neither proven nor disproved) to this day.
However, mathematicians have studied it intensively in two different aspects, analytical and numerical. The strongest result obtained analytically is the demonstration that at least 41.28% of the nontrivial zeros lie on the critical line; and the strongest result obtained numerically is the verification that all of the first ten trillion nontrivial zeros lie on the critical line online.