What question did this study set out to answer?

This research aims to explore how inversion symmetry influences the presence of zeros on the critical line of zeta-type functions.

May 16, 2026Open Access

Topological Forcing of Zeros on the Critical Line via Inversion Symmetry

Key Points

This research aims to explore how inversion symmetry influences the presence of zeros on the critical line of zeta-type functions.
Develops a cohomological obstruction mechanism to analyze zero avoidance on the critical line.
Uses winding classes from the first cohomology group in complex analysis to demonstrate symmetry constraints.
Applies the inversion map to show the relationship between the winding number and zero locations.
Winding classes in the first cohomology group indicated a necessity for zero winding number under inversion symmetry.
Curves with nonzero winding numbers cannot maintain inversion self-consistency without crossing the origin.
Demonstrates that topological contradictions can only be resolved by forcing zeros on the critical line.

Abstract

This paper formulates a cohomological obstruction mechanism where inversion symmetry forces a zero on the critical line by making "zero-avoidance" topologically impossible. The author establishes that if a de-symmetrized zeta-type function avoids the origin on the critical line, its image carries a winding class in the first cohomology group H¹ (C^; Z) Z. Because the inversion map (z) = z^-1 reverses this class (sending n to -n), a requirement for inversion self-consistency (=) forces the winding number to be zero. Consequently, if a curve possesses a nonzero winding number, it cannot satisfy this symmetry restoration while remaining in the punctured plane; the resulting topological contradiction can only be resolved if the curve crosses the origin, thereby necessitating a critical-line zero.

Topological Forcing of Zeros on the Critical Line via Inversion Symmetry

Key Points

Abstract

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