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

Abstract

This paper has two main results, which relate to a criteria for the Riemann hypothesis via the family of functions $\Theta_\omega(z)=\xi(1/2-\omega-iz)/\xi(1/2+\omega-iz)$, 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 $\Theta_\omega$ to be a meromorphic inner function in the upper half-plane. It is related to the Riemann hypothesis directly whether $\Theta_\omega$ 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 $\Theta_\omega$ 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.