Two complete and minimal systems associated with the zeros of the Riemann zeta function

Burnol, Jean-François

Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 65-94.

Nous relions trois thèmes restés jusqu’alors distincts : les propriétés hilbertiennes des zéros de Riemann, la “formule duale de Poisson” de Duffin-Weinberger (que nous appelons formule de co-Poisson), les espaces de fonctions entières “de Sonine” définis et étudiés par de Branges. Nous déterminons dans quels espaces de Sonine (étendus) les zéros forment un système complet, ou minimal. Nous obtenons des résultats généraux concernant la distribution des zéros des fonctions entières de de Branges-Sonine. Nous attirons l’attention sur certaines distributions liées à la transformation de Fourier et qui sont apparues dans nos travaux antérieurs.

We link together three themes which had remained separated so far: the Hilbert space properties of the Riemann zeros, the “dual Poisson formula” of Duffin-Weinberger (also named by us co-Poisson formula), and the “Sonine spaces” of entire functions defined and studied by de Branges. We determine in which (extended) Sonine spaces the zeros define a complete, or minimal, system. We obtain some general results dealing with the distribution of the zeros of the de-Branges-Sonine entire functions. We draw attention onto some distributions associated with the Fourier transform and which we introduced in our earlier works.

Mots clés : Riemann zeta function, Hilbert spaces, Fourier Transform

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