What question did this study set out to answer?

To explore the relationship between the Riemann zeta function and geometric structures that define its behavior.

March 14, 2026Open Access

Riemann Zeta Function and The Perfect Helix Regime

Key Points

To explore the relationship between the Riemann zeta function and geometric structures that define its behavior.
Introduced a geometric framework for Dirichlet partial sums of the Riemann zeta function.
Analyzed the remainder through second-stage projection onto a helical mode.
Investigated the dichotomy between line and wave states in the co-rotating helix frame.
Identified a perfect helix regime and its relation to wave states.
Demonstrated that the wave component cannot produce the vanishing of second analysis.
Characterized the zero condition through geometric collapse onto the perfect helix regime.

Abstract

We introduce a geometric framework for the Dirichlet partial sums of the Riemann zeta function based on a canonical helical carrier arising from the principal asymptotic term n^ (1/2-it). After subtracting the principal term, we analyze the remainder through a second-stage projection onto the helical mode. In the co-rotating helix frame this leads to a dichotomy between a line state (perfect helix regime) and a wave state. We show that the wave component coincides with the second-stage residual and cannot produce vanishing of the second analysis. This yields a geometric characterization of the zero condition in terms of collapse onto the perfect helix regime.

Riemann Zeta Function and The Perfect Helix Regime

Key Points

Abstract

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