On the Essence of the Riemann Zeta Function and Riemann Hypothesis

Riemann’s functional equation \(\pi^{- \frac{s}{2}}\Gamma\left( \frac{s}{2} \right)\zeta(s) = \pi^{- \left( \frac{1}{2} - \frac{s}{2} \right)}\Gamma\left( \frac{1}{2} - \frac{s}{2} \right)\zeta(1 - s)\) is valid on the vertical line \(s = \frac{1}{2} + it\). Each side is a real-valued function. The Riemann’s Xi function is also a real-valued function along the vertical line of \(s = \frac{1}{2} + it\). Through the holomorphic extensions of the Riemann zeta function, starting from the real-valued function at \(s = \frac{1}{2} + it\) into the both sides of \(\sigma < \frac{1}{2}\) and \(\sigma > \frac{1}{2}\), we can get two versions of the zeta functional equation, eq. (45). The key property of the scaling and rotational factors g(s) and g(1-s) behave as multiplicative inverses in the complex plane, eq. (48). It is deduced that the Zeta function also has multiplicative inverses, the symmetric point is at (1/2,0) in the complex plane. The moduli behave as a hyperbola. Especially, along the vertical line \(\sigma = \frac{1}{2} + it\), the amplitudes of both function g(s) and g(1-s) are equal to 1, its arguments have opposite signs. If \(\sigma \neq \frac{1}{2}\), the amplitudes of \(\zeta(s)\) and \(\zeta(1 - s)\) are not equal to each other, because of their multiplicative inversion relationship. It is deduced that the non-trivial zeros can only be on the vertical line of \(s = \frac{1}{2} + it\). A gamma function vector field is given in Appendix B, and some moduli of gamma function at special points are given. Finally, another variation of the Zeta function is provided in an integral form in Appendix D. The asymptotes behave as a c8 cyclic group for the large t values.