A canonical system of differential equations arising from the Riemann zeta-function

This paper has two main results, which relate to a criteria for the Riemann hypothesis via the family of functions ${\mathrm{\Theta }}_{\omega }(z)=\xi (1\mathrm{/}2-\omega -iz)\mathrm{/}\xi (1\mathrm{/}2+\omega -iz)$, where $\omega >0$ is a real parameter and $\xi (s)$ is the Riemann xi-function. The first main result is necessary and sufficient conditions for ${\mathrm{\Theta }}_{\omega }$ to be a meromorphic inner function in the upper half-plane. It is related to the Riemann hypothesis directly whether ${\mathrm{\Theta }}_{\omega }$ is a meromorphic inner function. In comparison with this, a relation of the Riemann hypothesis and the second main result is indirect. It relates to the theory of de Branges, which associates a meromorphic inner function and a canonical system of linear differential equations (in the sense of de Branges). As the second main result, the canonical system associated with ${\mathrm{\Theta }}_{\omega }$ is constructed explicitly and unconditionally under the restriction of the parameter $\omega >1$ by applying a method of J.-F. Burnol in his recent work on the gamma function to the Riemann xi-function. If such construction is extended to all $\omega >0$ unconditionally, we get a criterion for the Riemann hypothesis in terms of a family of canonical systems parametrized by $\omega >0$, which explains the validity of the Riemann hypothesis as positive semidefiniteness of the corresponding family of Hamiltonian matrices.