The Riemann Hypothesis and Beurling's theorem

Beurling's theorem about invariant subspaces in the Hardy spaces allows us to reformulate the Riemann Hypothesis about the zeros of the zeta function in the sense that a certain modification of the zeta function is an outer function. Unitary transplants of the construction into other Hilbert spaces give us equivalent reformulations of results for the Hardy class. Two such constructions will be considered. One of them is Beurling–Nyman's construction, and the other is connected with a formula established by Davenport.