Two Proofs of Riemann Hypothesis by Vector Properties of Riemann Zeta Function

The Riemann zeta function(RZF) ζ(s) is useful in number theory for studing properties of prime numbers. The Dirichlet eta function(DEF) η(s) is modification of RZF. In this thesis, we treat each term of RZF and DEF as a vector. From the geometric properties of vectors, we got clues of proof from the fact that, in a complex variable s = α + iβ, α only affects the magnitude of each vector and β affects only the argument of each vector, independently. So, each vector with same n are parallel to each other, regardless of the value of α. This parallel property implies a very strict geometric restriction which lead to two successful proofs of Riemann Hypothesis(RH). One proof is from the contradictions which come from the trajectories of RZF, and the other proof is by applying Chauchy integral theorem to the trajectory of RZF. We tried to provide sufficient graphs and videos for the understanding of the vector geometry properties of RZF and DEF. In appendix, we provided the source programs for analyzing vectors and suggested two other possible proofs of RH for further studies.