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 ∈ 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 its inverse 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.