Riemann hypothesis

The Riemann hypothesis is one of the most important and famous unsolved problems in number theory, proposed by German mathematician Bernhard Riemann in 1859. The conjecture concerns the Riemann zeta function, which is defined as:

[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} ]

Where the series converges and defines an analytical function for complex numbers (s) with real part greater than 1. By analytical extension, this function can be extended to the entire complex plane (except for a pole at (s = 1)).

The Riemann hypothesis is specifically stated as:

All non-trivial zeros have real part 1/2.

Non-trivial zeros are zeros of the zeta function other than (s = -2, -4, -6, \ldots) (these are called trivial zeros).

Importance

The importance of the Riemann hypothesis lies in its deep connection with the distribution of prime numbers. A prime number is a natural number greater than 1 that is divisible only by 1 and itself. The distribution of prime numbers is one of the core problems in number theory, and the Riemann hypothesis provides a way to understand the distribution of prime numbers. If the Riemann hypothesis is proved, it will provide more precise information about the distribution of prime numbers in natural numbers.

Related work

Since Riemann proposed this conjecture, mathematicians have conducted a lot of research and computational verification, confirming that many non-trivial zeros are indeed 1/2, but the complete proof has not yet been found.

Many famous mathematicians have tried to solve this problem, including David Hilbert and Alan Turing. So far, the number of zeros that have been verified has reached billions, but this does not prove that all zeros conform to this law.

Modern research

The Riemann hypothesis is still one of the hot spots in modern mathematical research. It is listed as one of the Millennium Prize Problems, and the Clay Mathematics Institute has offered a $1 million prize for it. Whether it is proved to be correct or wrong, the solution of the Riemann hypothesis will have a profound impact on number theory and many related fields.