Rigidity and Structural Asymmetry of Bounded Solutions

TL;DR

This paper studies a family of complex differential equations and shows that bounded solutions with specific initial conditions imply the real part of the parameter is 1/2, revealing a structural asymmetry related to the critical strip.

Contribution

It introduces a new 'Rotation number hypothesis' that links bounded solutions of parametrized differential equations to the critical line in the complex plane.

Findings

Bounded solutions with the same initial condition imply Re(s) = 1/2.

Establishes a structural asymmetry related to the critical strip.

Introduces the 'Rotation number hypothesis' for these equations.

Abstract

In this manuscript, we introduce a family of parametrized non-homogeneous linear complex differential equations on $[1,\infty)$, depending on a complex parameter. We identify a "Rotation number hypothesis" on the non-homogeneous term, which establishes a structural asymmetry: if two solutions with the same initial condition equal to $1$, corresponding respectively to the parameters $s$ and $1-\overline{s}$ lying in the critical strip, are both bounded on $[1,+\infty)$, then $\Re(s) = \tfrac{1}{2}$.