Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 3, pp. 621-664.

Nous rappelons quelques unes des principales propriétés d’universalité de la fonction zêta de Riemann ζ(s). De plus, nous expliquons comment obtenir une quantification naturelle du théorème d’universalité de Voronin (et de ses généralisations). Notre travail est basé sur la théorie des cordes fractales et de leurs dimensions complexes développée par le deuxième auteur et M. van Frankenhuijsen dans [60]. Nous utilisons également de façon essentielle la théorie développée dans [25] par les auteurs de cet article afin d’étudier de manière rigoureuse l’opérateur spectral (qui relie la géométrie et le spectre des cordes fractales généralisées). Cet opérateur spectral est representé (au sens du calcul fonctionnel) comme la composée de la fonction zêta de Riemann et du ‘shift infinitesimal’ (ou ‘décalage infinitésimal’) :𝔞=ζ(). Dans le processus de quantification du théorème d’universalité de la fonction zêta de Riemann, le rôle joué par la variable s (dans le théorème classique d’universalité) est joué par la famille des ‘shifts infinitésimaux tronqués’ introduite dans [25] afin d’étudier l’opérateur spectral en lien avec la reformulation spectrale de l’hypothèse de Riemann, vue comme un problème spectral inverse pour les cordes fractales. Ce dernier résultat fournit une version opératorielle de la reformulation spectrale obtenue par le second auteur et H. Maier dans [50]. Au long terme, notre présent travail (ainsi que [42, 43]), a en partie pour but d’obtenir une quantification naturelle de divers aspects de la théorie analytiques des nombres et de la géométrie arithmétique.

We survey some of the universality properties of the Riemann zeta function ζ(s) and then explain how to obtain a natural quantization of Voronin’s universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in [60]. It also makes an essential use of the functional analytic framework developed by the authors in [25] for rigorously studying the spectral operator 𝔞 (mapping the geometry onto the spectrum of generalized fractal strings), and the associated infinitesimal shift of the real line: 𝔞=ζ(), in the sense of the functional calculus. In the quantization (or operator-valued) version of the universality theorem for the Riemann zeta function ζ(s) proposed here, the role played by the complex variable s in the classical universality theorem is now played by the family of ‘truncated infinitesimal shifts’ introduced in [25] in order to study the invertibility of the spectral operator in connection with a spectral reformulation of the Riemann hypothesis as an inverse spectral problem for fractal strings. This latter work provided an operator-theoretic version of the spectral reformulation obtained by the second author and H. Maier in [50]. In the long term, our work (along with [42, 43]), is aimed in part at providing a natural quantization of various aspects of analytic number theory and arithmetic geometry.

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Herichi, Hafedh; Lapidus, Michel L. Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 3, pp. 621-664. doi : 10.5802/afst.1419. http://archive.numdam.org/articles/10.5802/afst.1419/

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