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We study the Hilbert space obtained by completing the space of all smooth andcompactly supported functions on the real line with respect to the hermitianform arising from the Weil distribution under the Riemann hypothesis. It turnsout that this Hilbert space is isomorphic to a de Branges space by acomposition of the Fourier transform and a simple map.This result is applied tostate a new equivalence condition for the Riemann hypothesis in a series ofequalities.

On the Hilbert space derived from the Weil distribution