Non-Trivial Zeros of the Riemann Zeta Function as Zero Displacement Vectors

Abstract

In this paper, we show that a non-trivial zero of the Riemann zeta function occurs only when the complex number s =  a/b + it, with a, b, t element of R and i² = -1 can be interpreted as a vector plus its inverse yielding zero displacement. We prove that for such a zero displacement to occur, the total distance covered by the vector and itsinverse must equal one unit, forcing the fundamental part of s to be 1 2. We further show that no other fraction in the critical strip possesses this property. Consequently, no other fundamental part can host non-trivial zeros, thereby settling the Riemann Hypothesis.