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This research aims to explore the geometric structure related to the zeros of the zeta function and their implications.

April 3, 2026Open Access

A Geometric Observation Concerning the Zeros of The Zeta Function

Key Points

This research aims to explore the geometric structure related to the zeros of the zeta function and their implications.
Constructed a six-vertex polytope with specific centroids
Interpreted the structure as both a regular octahedron and a sphere
Related discrete and continuous faces to zeta function properties
Developed a chain of implications linking independence and zero identification
Identified the explicit formula as a bridge between interpretations
Established Re(ρ) = 1/2 for all non-trivial zeros
Demonstrated connections to the linear independence conjecture and zero simplicity

Abstract

A six-vertex polytope is constructed whose centroid coincides with the critical line Re (s) = 1/2. The structure admits two simultaneous interpretations — a regular octahedron and a sphere — whose discrete and continuous faces correspond respectively to the Euler product and the functional equation of the Riemann zeta function. The Explicit Formula is identified as the bridge between these two faces. A chain of implications is developed from the Q-linear independence of log p through the aperiodicity of ψ (x), the Linear Independence Conjecture, zero simplicity, and the vertex identification, terminating in Re (ρ) = 1/2 for all non-trivial zeros.

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