Non-Trivial Zeros of the Riemann Zeta Function as Zero Displacement Vectors
Joseph Kongani Wamukoya
Joseph Kongani Wamukoya, Department of Math/Physics, Nairobi, Westlands, Nairobi, Kenya.Ā Ā Ā
Manuscript received on 07 January 2026 | First Revised Manuscript received on 15 January 2026 | Second Revised Manuscript received on 19 March 2026 | Manuscript Accepted on 15 April 2026 | Manuscript published on 30 April 2026Ā | PP: 9-10Ā | Volume-6 Issue-1, April 2026 | Retrieval Number: 100.1/ijam.A123306010426 | DOI: 10.54105/ijam.A1233.06010426
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Ā© The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Abstract: In this paper, we show that a non-trivial zero of the Riemann zeta function occurs only when the complex number s = š/š + it, with š, š, š ā š¹ and iĀ² = -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 š š . 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.
Keywords: Riemann Zeta Function, Equal One Unit, Fundamental Part.
Scope of the Article: Number Theory