On a summer day in 1900, more than 200 of the most outstanding mathematicians gathered in Paris, France, for an International Congress of Mathematicians. At the conference, the famous German mathematician Hilbert gave an important speech titled "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. A hundred years later, in 2000, mathematicians from the Clay Mathematics Institute in the United States also held a mathematics conference in Paris. At the conference, participants also listed some of what they considered to be the most important mathematical problems. Although their prestige might not be comparable to that of Hilbert, they did something that Hilbert could not do: set a huge prize of $1 million for each problem. Apart from both conferences being held in Paris, France, there is another striking similarity between them: among the listed problems, there is only one that they have in common. This problem is the Riemann Hypothesis, proposed by Riemann in 1859, which is regarded by many mathematicians as one of the most important mathematical conjectures.
What is the Riemann Hypothesis? The content of the Riemann Hypothesis cannot be fully described with elementary mathematics. Roughly speaking, it is a conjecture about a complex-valued function called the Riemann zeta function (i.e., a function whose variables and function values are all in the complex number field). Like many other functions, the Riemann zeta function takes the value of zero at certain points, which are called the zeros of the Riemann zeta function. Among those zeros, there are some particularly important ones called non-trivial zeros of the Riemann zeta function. The Riemann Hypothesis conjectures that all non-trivial zeros are distributed on a special line called the "critical line", which is a line parallel to the imaginary axis passing through the point 1/2 on the real axis.
The Riemann Hypothesis remains unresolved to this day (neither proven nor disproven). However, mathematicians have conducted in-depth research on it from two different perspectives: analysis and numerical calculation. So far, the strongest result obtained in the analysis is that at least 40% of the non-trivial zeros are located on the critical line; while the strongest result obtained in numerical calculation is that all of the first 100 trillion non-trivial zeros are located on the aforementioned critical line.
Why is the Riemann Hypothesis so important? Is it because it is extremely difficult? No. Of course, the Riemann Hypothesis is indeed very difficult, but if we only judge it by its difficulty, then other famous mathematical conjectures are no less difficult. For example, Fermat's Last Theorem was proven after more than 350 years of effort; the Goldbach Conjecture was proposed more than 100 years before the Riemann Hypothesis, but like the Riemann Hypothesis, it has stood unshaken to this day. So why is the Riemann Hypothesis called the most important mathematical conjecture? The first reason is that it is closely related to other mathematical propositions. According to statistics, there are more than 1,000 mathematical propositions in today's mathematical literature that are based on the establishment of the Riemann Hypothesis (or its generalized form). This shows that once the Riemann Hypothesis and its generalized form are proven, all of those more than 1,000 mathematical propositions can be elevated to theorems; conversely, if the Riemann Hypothesis is disproven, at least some of those more than 1,000 mathematical propositions will inevitably become "sacrifices". It is almost unique in mathematics for a mathematical conjecture to be so closely related to so many mathematical propositions. Secondly, 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, which was called the "queen of mathematics" by the German mathematician Gauss. The distribution of prime numbers is also an extremely important traditional topic in number theory, which has always attracted the interest of many mathematicians. This has greatly increased the importance of the Riemann Hypothesis in the eyes of mathematicians.The status and importance of the Riemann Hypothesis in the eyes of mathematicians. Furthermore, another criterion for measuring the importance of a mathematical conjecture is whether it can lead to methods and results that contribute to and influence other aspects of mathematics during the process of studying the conjecture. In fact, one of the early achievements made by mathematicians in studying the Riemann Hypothesis directly led to the proof of an important proposition on the distribution of prime numbers—the Prime Number Theorem. Finally, and most surprisingly, the importance of the Riemann Hypothesis even extends beyond the realm of pure mathematics and "invades" the field of physics. In the early 1970s, it was discovered that some research related to the Riemann Hypothesis actually had significant correlations with certain highly complex physical phenomena. The reason for this correlation remains a mystery to this day, but its existence undoubtedly further enhances the importance of the Riemann Hypothesis.