Shifted Resonance Horizontal Criterion

Abstract

The shifted zero points lie on one horizontal line exactly when every zero has critical real part.

Theorem 1.1 (Shifted resonances characterize the critical line).

$$\forall Z:\mathrm{ZeroData},\forall omega:\mathbb{R},\frac{1}{2}\le omega\Rightarrow \left(\left(\forall n\in \mathbb{N},\Re (Z.zero(n))=\mathrm{criticalAbscissa}\left(\right)\right)\iff \mathrm{rangeIntersectUpper}\left(\mathrm{resonance}\left(omega\right)\right)\subseteq \mathrm{horizontalLine}\left(omega\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Weil/Scattering/ShiftedResonanceCriterion.horizontal_resonance_line_iff_critical_line (✓ std3). ∎

Source. Repository-derived.

Commentary.

For supplied ZeroData and a real shift omega at least one half, the public resonance map sends index n to the complex point minus Im(Z.zero n) plus i times omega plus Re(Z.zero n) minus criticalAbscissa. The displayed inclusion is the range of this map intersected with the upper half-plane contained in the horizontal line of height omega.

Every enumerated zero is in the open strip, so the shifted points are automatically in the upper half-plane for the stated shift. The two directions then reduce the line condition to equality of each zero real part with the critical abscissa.

References

• Truth anchor: D5/S3/Weil/Scattering/ShiftedResonanceCriterion.horizontal_resonance_line_iff_critical_line