ζ(s)= 2γ( 1-s)(2π)s- 1 sin(πs/2)ζ( 1-s)
From this relation, it is not difficult to find that the Riemann zeta function takes zero at s=-2n (n is a positive integer)-because sin(πs/2) is zero [Note 3]. The point on the complex plane that makes the value of Riemann zeta function zero is called the zero point of Riemann zeta function. So s=-2n (n is a positive integer) is the zero point of the Riemann zeta function. These zeros are ordered in distribution and simple in nature, so they are called ordinary zeros of Riemann zeta function. In addition to these trivial zeros, Riemannian zeta function has many other zeros, and their properties are far more complicated than trivial zeros. They are called nontrivial zeros. The study of nontrivial zeros of Riemannian zeta function is one of the most difficult topics in modern mathematics. Riemann conjecture is a conjecture about these nontrivial zeros.
Riemann conjecture: All nontrivial zeros of Riemann zeta function lie on the straight line of Re(s)= 1/2 on the complex plane.
This is the content of Riemann conjecture, which was put forward by Riemann in 1859. Riemann conjecture seems to be a proposition of pure complex variable function in terms of expression, but it is actually a mysterious movement about the distribution of prime numbers.
An attempt to prove riemann conjecture
Riemann 1859 in his thesis &; The term of office of the first prime minister of Uuml is one year. Ouml& ampszlige' mentioned this famous conjecture, but this is not the central purpose of this article, and he has not tried to prove it. Riemann knows that the nontrivial zeros of zeta function are symmetrically distributed on the straight line S =&; frac 12； +it, and he knows that all its very zeros must be in the region of 0 ≤ Re(s) ≤ 1.
1896, Jacques Adama and Charles Jean de la Valle-Pu Sang independently proved that there is no zero point on the straight line Re(s) = 1. Together with other characteristics of extraordinary zeros that Riemann has proved, it shows that all extraordinary zeros must be in region 0.
1900, david hilbert included Riemann conjecture in his famous 23 questions, and Riemann conjecture and Goldbach conjecture together constituted the eighth question on Hilbert table. When asked what he would do if he woke up 500 years later, Hilbert famously said that his first question would be whether the Riemann conjecture was proved. (Derbyshire 2003:197; Sabbag 2003: 69; Bollobas 1986: 16)。 Riemann conjecture is the only Hilbert problem included in the Millennium Prize of Clay Institute of Mathematics.
In 19 14, Godfrey Harold Hardy proved that there are infinite zeros in the straight line Re(s) = & frac 12; Let's go However, it is still possible to have an infinite number of very zeros (perhaps the most important one) elsewhere. Later, Hardy and John Edensor Littlewood's work in 192 1 (critical line theorem) and Selberg's work in 1942 were to calculate the zero point Re (s) = & frac12 on the critical line; The average density of.
In recent decades, the focus of work has been to clearly calculate the positions of a large number of zeros (hoping to find counterexamples) and set an upper limit on the proportion of zeros outside the critical line (hoping to reduce the upper limit to zero).
In the past decades, many mathematicians claimed to have proved the Riemann conjecture, but as of 2007, there were still a few proofs that had not been verified. But they are all questioned by the mathematical community, and most experts don't believe that they are correct. Matthew R. Watkins of the University of Eicht compiled a list of these serious or absurd statements, and some other proofs of these statements can be found in the arXiv database.