A Rigidity Hypothesis for the Structural Asymmetry of the Bounded Solutions of a Parametric Complex Linear Differential Equation

In the equality $=0$ in the proof of Proposition~7, there is a missing term, namely $\sqrt{t}-1$; therefore this term must be removed from the equalities that follow. The proof remains valid. This is also shown by the numerical results, which validate hypothesis $H$ when $\eta$ is the fractional part. The coefficient of $\ln(t)$ in $f_\beta$, or of $\sqrt{t}$ in $h_\beta$, must be non-negative as $t \to +\infty$ for the proof to be valid. Numerical tests show that it is indeed non-negative. The coefficient at infinity is nothing but the real part of the function \[ \beta \mapsto -\frac{1-s}{s}\zeta(s) \quad \text{with} \quad s := \frac{1}{2} + i\beta. \] It therefore vanishes at the non-trivial zeros of the function, and this suggests that the preceding function might be written in the form \[ 1 - \cos(\Phi(\beta)) \quad \text{or} \quad 1 - \sin(\Phi(\beta)), \] where the real function $\Phi$ is an increasing function of the single variable $\beta$ and tends to $+\infty$.