Riemann observed that the frequency of prime numbers is closely related to the behavior of the so-called Riemann zeta function zeta (S). Riemann hypothesis asserts that all meaningful solutions of equation ζ(s)=0 are on a straight line. This has been verified in the original 1, 500,000,000 solutions.

Riemann zeta function zeta (s) is a series expression [8]

Analytic continuation on complex plane.

The reason why we want to generalize this expression analytically is that it only applies to the real part of S, the region of Re(S)>: 1 on the complex plane (otherwise the series will not converge). Riemann found the analytic extension of this expression (of course, Riemann did not use the modern terms of complex variable function theory such as "analytic extension"). Using path integral, the analytic extended Riemann zeta function can be expressed as [8]

Revealing the Truth of zeta Function in Riemann Manuscripts. Baidu Library.2015-08-16 [citation date 2015-19]

Riemannian geometry is a kind of non-Euclidean geometry, also called "elliptic geometry". German mathematician Riemann made a deep research on the concepts of space and geometry. 1854 published the article "On Hypothesis as the Basis of Geometry" and founded Riemann Geometry.

A basic law in Riemannian geometry is that any two straight lines on the same plane have common points (intersections). In Riemannian geometry, the existence of parallel lines is not recognized, and its other postulate says that straight lines can extend indefinitely, but the total length is limited. The model of Riemannian geometry is a sphere that has been properly "improved".

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5. On the number of prime numbers less than a given value (Riemann's original paper on Riemann conjecture)-translated by Xie Guofang. Digital Light [citation date: 20 15-08-2 1]