Charles Ferreira dos Santos (University of São Paulo) : Density and orthogonality in a Hardy space reformulation of Riemann hypothesis

Séminaire « Analyse fonctionnelle »
Salle Kampé de Fériet, Bâtiment M2

Lieu : Salle Kampé de Fériet, 1er étage, Bâtiment M2

Date : 7 juin 2024

Heure : 14h

Orateur : Charles dos Santos

Affiliation : University of São Paulo

Résumé : Báez-Duarte’s criterion asserts that the Riemann hypothesis (RH) is equivalent to the density of the linear span of a particular sequence in L²([0, 1]). This talk concerns a unitarily equivalent
version in the classical Hardy space H² of the unit disk: RH is true if and only if a certain subspace N is dense in H². This poses 2 immediate questions: (i) How to weaken the topology in order to assure
the density of N? (ii) What elements of H² can be orthogonal to N? For question (i), N was shown to be dense in Hp for p<1, and density for some p>1 would give a non-trivial zero-free half plane for the
Riemann-zeta function. Density in H¹ remains open. Question (ii) can be given answers involving spaces of Hölder continuous functions up to the unit circle and de Branges-Rovnyak spaces. Roughly, a non null function orthogonal to N must be irregular at the boundary. This is a joint work with Waleed Noor (IMECC/Unicamp), A. Ghosh and K. Kremnizer (Oxford University, UK).

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