On the Hilbert space derived from the Weil distribution

TL;DR

This paper investigates a Hilbert space constructed from the Weil distribution under the Riemann hypothesis, revealing its isomorphism to a de Branges space and proposing a new equivalence condition for the hypothesis.

Contribution

It establishes an isomorphism between the Weil distribution-derived Hilbert space and a de Branges space, offering a novel perspective on the Riemann hypothesis.

Findings

Hilbert space from Weil distribution is isomorphic to a de Branges space

New equivalence condition for the Riemann hypothesis

Connection between Fourier transform and space structure

Abstract

We study the Hilbert space obtained by completing the space of all smooth and compactly supported functions on the real line with respect to the hermitian form arising from the Weil distribution under the Riemann hypothesis. It turns out that this Hilbert space is isomorphic to a de Branges space by a composition of the Fourier transform and a simple map.This result is applied to state a new equivalence condition for the Riemann hypothesis in a series of equalities.