This paper has two main results, which relate to a criteria for the Riemann
hypothesis via the family of functions
Θω​(z)=ξ(1/2−ω−iz)/ξ(1/2+ω−iz), where ω>0 is a
real parameter and ξ(s) is the Riemann xi-function. The first main result
is necessary and sufficient conditions for Θω​ to be a meromorphic
inner function in the upper half-plane. It is related to the Riemann hypothesis
directly whether Θω​ 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 Θω​ is constructed explicitly and
unconditionally under the restriction of the parameter ω>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 ω>0
unconditionally, we get a criterion for the Riemann hypothesis in terms of a
family of canonical systems parametrized by ω>0, which explains the
validity of the Riemann hypothesis as positive semidefiniteness of the
corresponding family of Hamiltonian matrices.Comment: 28 pages, draft of a paper which will be published in a volume of
RIMS Kokyuroku Bessatsu Serie