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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.

@article{AFST_2014_6_23_3_621_0,
     author = {Hafedh Herichi and Michel L. Lapidus},
     title = {Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {621--664},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 23},
     number = {3},
     year = {2014},
     doi = {10.5802/afst.1419},
     zbl = {06374883},
     mrnumber = {3266708},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2014_6_23_3_621_0/}
}
Hafedh Herichi; Michel L. Lapidus. 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. https://afst.centre-mersenne.org/item/AFST_2014_6_23_3_621_0/

[1] Bagchi (B.).— The statistical behaviour and universality of the Riemann zeta function and other Dirichlet series, Ph.D. Thesis, Indian Statistical Institute, Calcutta, India (1981).

[2] Bagchi (B.).— A joint universality theorem for Dirichlet L-functions, Math. Z. 181, p. 319-334 (1982). | | MR 678888 | Zbl 0479.10028

[3] Besicovitch (A. S.) and Taylor (S. J.).— On the complementary intervals of a linear closed set of zero Lesbegue measure, J. London Math. Soc. 29, p. 449-459 (1954). | MR 64849 | Zbl 0056.27801

[4] Bitar (K. M.), Khuri (N. N.), Ren (H. C.).— Path integrals and Voronin’s theorem on the universality of the Riemann zeta function, Ann. Phys. 211 (1), p. 151-175 (1991). | MR 1128186 | Zbl 0764.11055

[5] Bohr (H.).— Zur Theorie der Riemannschen ZetaFunktion im kritischen Streifen, Acta Math. 40, p. 67-100 (1915). | JFM 45.0719.01 | MR 1555133

[6] Bohr (H.).— Über eine quasi-periodische Eigenschaft Dirichletscher Reihen mit Anwendung auf die Dirichletschen L-Funktionen, Math. Ann. 85, p. 115-122 (1922). | | JFM 48.0343.02 | MR 1512052

[7] Bohr (H.) and Courant (R.).— Neue Anwendungen der Theorie der diophantischen Approximationen auf die Riemannsche Zetafunktion, J. Reine Angew. Math. 144, p. 249-274 (1914). | | JFM 45.0718.02

[8] Brezis (H.).— Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. (English transl. and rev. and enl. edn. of H. Brezis, Analyse Fonctionelle: Théorie et applications, Masson, Paris (1983).) | MR 2759829 | Zbl 1220.46002

[9] Cohn (D. L.).— Measure Theory, Birkhäuser, Boston, 1980. | MR 578344 | Zbl 0860.28001

[10] Dunford (N.) and Schwartz (J. T.).— Linear Operators, Parts I-III, Wiley Classics Library, John Wiley & Sons, Hoboken (1988). (Part I: General Theory. Part II: Spectral Theory. Part III: Spectral Operators.) | Zbl 0635.47001

[11] Edwards (H. M.).— Riemann’s Zeta Function, Academic Press, New York (1974). (Paperback and reprinted edition, Dover Publications, Mineola, 2001.) | MR 1854455 | Zbl 1113.11303

[12] Ellis (K. E.), Lapidus (M. L.), Mackenzie (M. C.) and Rock (J. A.).— Partition zeta functions, multifractal spectra, and tapestries of complex dimensions, in: Benoit Mandelbrot: A Life in Many Dimensions, the Mandelbrot Memorial Volume (Frame (M.), (ed.)), World Scientific, Singapore, 2014, in press. (Also: e-print, arXiv:1007.1467v2 [math-ph], 2011; IHES preprint, IHES/M/12/15, 2012.)

[13] Eminyan (K. M.).— χ-universality of the Dirichlet L-function, Mat. Zametki 47 (1990), p. 132-137 (Russian); translation in Math. Notes 47, 618-622 (1990). | MR 1074538 | Zbl 0713.11058

[14] Engel (K.-J.) and Nagel (R.).— One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer, Berlin (2000). | MR 1721989 | Zbl 0952.47036

[15] Falconer (K. J.).— Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Chichester, (1990). (2nd edn., 2003.) | MR 2118797 | Zbl 0871.28009

[16] Folland (G. B.).— Real Analysis: Modern Techniques and Their Applications, 2nd edn., John Wiley & Sons, Boston (1999). | MR 1681462 | Zbl 0549.28001

[17] Garunkštis (R.).— The effective universality theorem for the Riemann zeta-function, in: Special Activity in Analytic Number Theory and Diophantine Equations, Proceedings of a workshop held at the Max Planck-Institut Bonn, 2002 (Heath-Brown (R. B.) and Moroz (B.), eds.), Bonner Math. Schriften 360 (2003). | MR 2075625 | Zbl 1070.11035

[18] Garunkštis (R.) and Steuding (J.).— On the roots of the equation ζ(s)=α, e-print, arXiv:1011.5339 [mathNT] (2010).

[19] Gauthier (P. M.) and Clouatre (R.).— Approximation by translates of Taylor polynomials of the Riemann zeta function, Computational Methods and Function Theory 8, p. 15-19 (2008). | MR 2419456 | Zbl 1221.30084

[20] Gonek (S. M.).— Analytic Properties of Zeta and L-functions, Ph. D. Thesis, University of Michigan, Ann Arbor (1979). | MR 2628587

[21] Good (A.).— On the distribution of the values of the Riemann zeta-function, Acta Arith. 38, p. 347-388 (1981). | MR 621007 | Zbl 0372.10029

[22] Haase (M.).— The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, vol. 169, Birkhäuser Verlag, Berlin (2006). | MR 2244037 | Zbl 1101.47010

[23] Hambly (B. M.) and Lapidus (M. L.).— Random fractal strings: their zeta functions, complex dimensions and spectral asymptotics, Trans. Amer. Math. Soc. No. 1, 358, p. 285-314 (2006). | MR 2171234 | Zbl 1079.60019

[24] He (C. Q.) and Lapidus (M. L.).— Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta-function, Memoirs Amer. Math. Soc. No. 608, 127, p. 1-97 (1997). | MR 1376743 | Zbl 0877.35086

[25] Herichi (H.) and Lapidus (M. L.).— Fractal Strings, Quantized Number Theory and the Riemann Hypothesis: From Infinitesimal Shifts and Spectral Operators to Phase Transitions and Universality, research monograph, preprint, (2014), approx. 170 pages.

[26] Herichi (H.) and Lapidus (M. L.).— Riemann zeros and phase transitions via the spectral operator on fractal strings, J. Phys. A: Math. Theor. 45 (2012) 374005, 23pp. (Also: e-print, arXiv:1203.4828v2 [math-ph], 2012; IHES preprint, IHES/M/12/09, 2012.) | MR 2970522 | Zbl 1252.81067

[27] Herichi (H.) and Lapidus (M. L.).— Fractal complex dimensions, Riemann hypothesis and invertibility of the spectral operator, in: Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I: Fractals in Pure Mathematics (Carfi (D.), Lapidus (M. L.), Pearse (E. P. J.) and van Frankenhuijsen (M.), eds.), Contemporary Mathematics, vol. 600, Amer. Math. Soc., Providence, R. I., 2013, p. 51-89. (Also: e-print, arXiv:1210.0882v3 [math.FA], 2012; IHES preprint, IHES/M/12/25, 2012.) | MR 3203399

[28] Herichi (H.) and Lapidus (M. L.).— Quantized Riemann zeta function: Its operator-valued Dirichlet series, Euler product and analytic continuation, in preparation (2014).

[29] Hille (E.) and Phillips (R. S.).— Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ., vol. XXXI, rev. edn., Amer. Math. Soc., R. I. (1957). | MR 89373 | Zbl 0078.10004

[30] Ingham (A. E.).— The Distribution of Prime Numbers, 2nd edn. (reprinted from the 1932 edn.), Cambridge Univ. Press, Cambridge (1992). | MR 1074573 | Zbl 0715.11045

[31] Ivic (A.).— The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function with Applications, John Wiley & Sons, New York (1985). | MR 792089 | Zbl 0556.10026

[32] Johnson (G. W.) and Lapidus (M. L.).— The Feynman Integral and Feynman’s Operational Calculus, Oxford Science Publications, Oxford Mathematical Monographs, Oxford Univ. Press, Oxford and New York, 2000. (Paperback edition and corrected reprinting (2002).) | MR 1771173 | Zbl 1027.46002

[33] Kac (M.).— Can one hear the shape of a drum?, Amer. Math. Monthly (Slaught Memorial Papers, No. 11) (4) 73, p. 1-23 (1966). | MR 201237 | Zbl 0139.05603

[34] Karatsuba (A. A.) and Voronin (S. M.).— The Riemann Zeta-Function, Expositions in Mathematics, Walter de Gruyter, Berlin (1992). | MR 1183467 | Zbl 0756.11022

[35] Kato (T.).— Perturbation Theory for Linear Operators, Springer-Verlag, New York (1995). | MR 1335452 | Zbl 0435.47001

[36] Lal (N.) and Lapidus (M. L.).— Hyperfunctions and spectral zeta functions of Laplacians on self-similar fractals, J. Phys. A: Math. Theor. 45 (2012) 365205, 14pp. (Also, e-print, arXiv: 1202.4126v2 [math-ph], 2012; IHES preprint, IHES/M/12/14, 2012.) | MR 2967908 | Zbl 1256.28004

[37] Lal (N.) and Lapidus (M. L.).— The decimation method for Laplacians on fractals: Spectra and complex dynamics, in: Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics (Carfi (D.), Lapidus (M. L.), Pearse (E. P. J.) and van Frankenhuijsen (M.), eds.), Contemporary Mathematics, vol. 601, Amer. Math. Soc., Providence, R. I., 2014, p. 227-249. (Also: e-print, arXiv: 1302.4007v2 [math-ph], 2014; IHES preprint, IHES/M/12/31, 2012.) | MR 3203865

[38] Lapidus (M. L.).— Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc. 325, p. 465-529 (1991). | MR 994168 | Zbl 0741.35048

[39] Lapidus (M. L.).— Spectral and fractal geometry: From the Weyl-Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function, in: Differential Equations and Mathematical Physics (Bennewitz (C.), ed.), Proc. Fourth UAB Internat. Conf. (Birmingham, March 1990), Academic Press, New York, p. 151-182 (1992). | MR 1126694 | Zbl 0736.58040

[40] Lapidus (M. L.).— Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl-Berry conjecture, in: Ordinary and Partial Differential Equations (Sleeman (B. D.) and Jarvis (R. J.), eds.), vol. IV, Proc. Twelfth Internat. Conf. (Dundee, Scotland, UK, June 1992), Pitman Research Notes in Math. Series, vol. 289, Longman Scientific and Technical, London, p. 126-209 (1993). | MR 1234502 | Zbl 0830.35094

[41] Lapidus (M. L.).— Fractals and vibrations: Can you hear the shape of a fractal drum?, Fractals No. 4, 3, p. 725-736 (1995). (Special issue in honor of Benoit B. Mandelbrot’s 70th birthday.) | MR 1410291 | Zbl 0870.58063

[42] Lapidus (M. L.).— In Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes, Amer. Math. Soc., Providence, R. I. (2008). | MR 2375028 | Zbl 1150.11003

[43] Lapidus (M. L.).— Quantized Weil conjectures, spectral operator and Polya-Hilbert operators (tentative title), in preparation (2014).

[44] Lapidus (M. L.), Lévy-Véhel (J.) and Rock (J. A.).— Fractal strings and multifractal zeta functions, Lett. Math. Phys. No. 1, 88, p. 101-129 (2009) (special issue dedicated to the memory of Moshe Flato). (Springer Open Acess: DOI 10.1007/s1105-009-0302-y.) (Also: e-print, arXiv:math-ph/0610015v3, 2009.) | MR 2512142 | Zbl 1170.11030

[45] Lapidus (M. L.) and Lu (H.).— Self-similar p-adic fractal strings and their complex dimensions, p-Adic Numbers, Ultrametric Analysis and Applications (Russian Academy of Sciences, Moscow, and Springer-Verlag), No. 2, 1, p. 167-180 (2009). (Also: IHES preprint, IHES/M/08/42, 2008.) | MR 2566062 | Zbl 1187.28014

[46] Lapidus (M. L.) and Lu (H.).— The geometry of p-adic fractal strings: A comparative survey, in: Advances in Non-Archimedean Analysis, Proc. 11th Internat. Conference on p-Adic Functional Analysis (Clermont-Ferrand, France, July 2010), Araujo (J.), Diarra (B.) and Escassut (A.), eds., Contemporary Mathematics, vol. 551, Amer. Math. Soc., Providence, R. I., 2011, p. 163-206. (Also: e-print, arXiv:1105.2966v1 [math.MG], 2011.) | MR 2882397 | Zbl 1276.37053

[47] Lapidus (M. L.), Lu (H.) and van Frankenhuijsen (M.).— Minkowski measurability and exact fractal tube formulas for p-adic self-similar strings, in: Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I: Fractals in Pure Mathematics (Carfi (D.), Lapidus (M. L.), Pearse (E. P. J.) and van Frankenhuijsen (M.), eds.), Contemporary Mathematics, vol. 600, Amer. Math. Soc., Providence, R. I., 2013, p. 161-184. (Also: e-print, arXiv:1209.6440v1 [math.MG], 2012; IHES preprint, IHES/M/12/23, 2012.) | MR 3203402 | Zbl 1276.00022

[48] Lapidus (M. L.), Lu (H.) and van Frankenhuijsen (M.).— Minkowski dimension and explicit tube formulas for p-adic fractal strings, preprint (2014).

[49] Lapidus (M. L.) and Maier (H.).— Hypothèse de Riemann, cordes fractales vibrantes et conjecture de Weyl-Berry modifiée, C. R. Acad. Sci. Paris Sér. I Math. 313, p. 19-24 (1991). | MR 1115940 | Zbl 0751.35030

[50] Lapidus (M. L.) and Maier (H.).— The Riemann hypothesis and inverse spectral problems for fractal strings, J. London Math. Soc. (2) 52, p. 15-34 (1995). | MR 1345711 | Zbl 0836.11031

[51] Lapidus (M. L.) and Pearse (E. P. J.).— Tube formulas and complex dimensions of self-similar tilings, Acta Applicandae Mathematicae No. 1, 112 (2010), p. 91-137. (Also: e-print, arXiv: math.DS/0605527v5, 2010; Springer Open Access: DOI 10.1007/S10440-010-9562-x.) | MR 2684976 | Zbl 1244.28013

[52] Lapidus (M. L.), Pearse (E. P. J.) and Winter (S.).— Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators, Adv. Math. 227 (2011), p. 1349-1398. (Also: e-print, arXiv:1006.3807v3 [math.MG], 2011.) | MR 2799798 | Zbl 1274.28016

[53] Lapidus (M. L.) and Pomerance (C.).— Fonction zêta de Riemann et conjecture de Weyl-Berry pour les tambours fractals, C. R. Acad. Sci. Paris Sér. I Math. 310, p. 343-348 (1990). | MR 1046509 | Zbl 0707.58046

[54] Lapidus (M. L.) and Pomerance (C.).— The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums, Proc. London Math. Soc. (3) 66, p. 41-69 (1993). | MR 1189091 | Zbl 0739.34065

[55] Lapidus (M. L.) and Pomerance (C.).— Counterexamples to the modified Weyl-Berry conjecture on fractal drums, Math. Proc. Cambridge Philos. Soc. 119, p. 167-178 (1996). | MR 1356166 | Zbl 0858.58052

[56] Lapidus (M. L.), Radunović (G.) and Z ˇubrinić (D.).— Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions, research monograph, preprint, 2014, approx. 335 pages.

[57] Lapidus (M. L.) and van Frankenhuijsen (M.).— Complex dimensions of fractal strings and oscillatory phenomena in fractal geometry and arithmetic, in: Spectral Problems in Geometry and Arithmetic (Branson (T.), ed.), Contemporary Mathematics, vol. 237, Amer. Math. Soc., Providence, R. I., p. 87-105 (1999). | MR 1710790 | Zbl 0945.11016

[58] Lapidus (M. L.) and van Frankenhuijsen (M.).— Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions, Birkhäuser, Boston (2000). | MR 1726744 | Zbl 0981.28005

[59] Lapidus (M. L.) and van Frankenhuijsen (M.).— Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, Springer Monographs in Mathematics, Springer, New York (2006). | MR 2245559 | Zbl 1119.28005

[60] Lapidus (M. L.) and van Frankenhuijsen (M.).— Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, second rev. and enl. edn. (of the 2006 edn., [59]), Springer Monographs in Mathematics, Springer, New York (2013). | MR 2245559 | Zbl 1261.28011

[61] Lapidus (M. L.) and van Frankenhuijsen (M.) (eds.).— Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Proc. Sympos. Pure Math., vol. 72, Parts 1 & 2, Amer. Math Soc., Providence, R. I. (2004).

[62] Laurinčikas (A.).— Limit Theorems for the Riemann Zeta-Function, Kluwer Academic Publishers, Dordrecht (1996). | MR 1376140 | Zbl 0845.11002

[63] Laurincikas (A.).— The universality of the Lerch zeta-function, Liet. Mat. Rink. 37 (1997), p. 367-375 (Russian); Lith. Math. J. 37, p. 275-280 (1997). | MR 1481388 | Zbl 0938.11045

[64] Laurincikas (A.).— Prehistory of the Voronin universality theorem, Siauliai Math. J. 1 (9), p. 41-53 (2006). | MR 2547354 | Zbl 1126.11041

[65] Laurincikas (A.) and Matsumoto (K.).— The joint universality of twisted automorphic L-functions, J. Math. Soc. Japan 56, p. 923-939 (2004). | MR 2071679 | Zbl 1142.11032

[66] Laurincikas (A.) and Matsumoto (K.).— The universality of zeta-functions attached to certain cusp forms, Acta Arith. 98, p. 345-359 (2001). | MR 1829777 | Zbl 0974.11018

[67] Laurincikas (A.), Matsumoto (K.) and Steuding (J.).— The universality of L-functions associated with newforms, Izvestija Math. 67 (2003), p. 77-90; Izvestija Ross. Akad. Nauk Ser. Mat. 67, p. 83-96 (2003) (Russian). | MR 1957917 | Zbl 1112.11026

[68] Laurincikas (A.) and Slezeviciene (R.).— The universality of zeta-functions with multiplicative coefficients, Integral Transforms and Special Functions 13, p. 243-257 (2002). | MR 1919181 | Zbl 1020.11057

[69] Laurincikas (A.) and Steuding (J.).— Joint universality for L-functions attached to a family of elliptic curves, in: Proceedings of the ELAZ 2004, Conference on “Elementary and Analytic Number Theory”, Mainz, 2004 (Schwartz (W.) and Steuding (J.), eds.), Steiner Verlag, Stuttgart, p. 153-163 (2006). | MR 2310179 | Zbl 1155.11033

[70] Mandelbrot (B. B.).— The Fractal Geometry of Nature, rev. and enl. edn. (of the 1977 edn.), W. H. Freeman, New York (1983). | MR 665254 | Zbl 0504.28001

[71] Mattila (P.).— Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge Univ. Press, Cambridge (1995). | MR 1333890 | Zbl 0911.28005

[72] Montgomery (H. L.).— Extreme values of the Riemann zeta-function, Comment. Math. Helv. 52, p. 511-518 (1997). | MR 460255 | Zbl 0373.10024

[73] Ostrowski (A.).— Über Dirichletsche Reihen und algebraische Differentialgleichungen, Math. Z. 8, 241-298 (1920). | MR 1544442

[74] Patterson (S. J.).— An Introduction to the Theory of the Riemann Zeta-Function, Cambridge Univ. Press, Cambridge (1988). | MR 933558 | Zbl 0831.11045

[75] Pazy (A.).— Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin and New York (1983). | MR 710486 | Zbl 0516.47023

[76] Pérez (D.) and Quintana (Y.).— A survey on the Weierstrass approximation theorem, Divulgaciones Mátematicas, No. 1, 16, p. 231-247 (2008). (Also: e-print, arXiv:math/0611038v2 [math.CA], 2008.) | MR 2587018 | Zbl 1217.41023

[77] Reich (A.).— Wertverteilung von Zetafunktionen, Arch. Math. 34, p. 440-451 (1980). | MR 593771 | Zbl 0431.10025

[78] Reich (A.).— Universelle Wertevereteilung von Eulerprodukten, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, Nos. 1-17 (1977). | MR 567687 | Zbl 0379.10025

[79] Reed (M.) and Simon (B.).— Methods of Modern Mathematical Physics, vol. I, Functional Analysis, rev. and enl. edn. (of the 1975 edn.), and vol. II, Fourier Analysis, Self-Adjointness, Academic Press, New York, 1980 and 1975. | MR 751959 | Zbl 0308.47002

[80] Riemann (B.).— Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsb. der Berliner Akad. 1858/60, pp. 671-680. (English transl. in [11, Appendix, p. 229-305].)

[81] Rudin (W.).— Functional Analysis, 2nd edn. (of the 1973 edn.), McGraw-Hill, New York (1991). | MR 1157815 | Zbl 0867.46001

[82] Schechter (M.).— Operator Methods in Quantum Mechanics, Dover Publications, New York (2003). | MR 1969612 | Zbl 1029.47054

[83] Schwartz (L.).— Théorie des Distributions, rev. and enl. edn. (of the 1951 edn.), Hermann, Paris (1996). | MR 41345 | Zbl 0042.11405

[84] Serre (J.-P.).— A Course in Arithmetic, English transl., Springer-Verlag, Berlin (1973). | MR 344216 | Zbl 0432.10001

[85] Steuding (J.).— Value-Distribution of L-Functions, Lecture Notes in Mathematics, vol. 1877, Springer, Berlin (2007). | MR 2330696 | Zbl 1130.11044

[86] Steuding (J.).— Universality in the Selberg class, in: Special Activity in Analytic Number Theory and Diophantine Equations, Proceedings of a workshop held at the Max Planck-Institut Bonn (2002) (Heath-Brown (R. B.) and Moroz (B.), eds.), Bonner Math. Schriften 360 (2003). | MR 2075637 | Zbl 1059.11052

[87] Teplyaev (A.).— Spectral zeta functions of fractals and the complex dynamics of polynomials, Trans. Amer. Math. Soc. 359, p. 4339-4358 (2007). | MR 2309188 | Zbl 1129.28010

[88] Titchmarsh (E. C.).— The Theory of the Riemann Zeta-Function, 2nd edn. (revised by Heath-Brown (D. R.)), Oxford Science Publications, Oxford Mathematical Monographs, Oxford Univ. Press, Oxford (1986). | MR 882550 | Zbl 0601.10026

[89] Voronin (S. M.).— The distribution of the non-zero values of the Riemann zeta function, Izv. Akad. Nauk. Inst. Steklov 128, p. 131-150 (in Russian) (1972). | MR 319915 | Zbl 0294.10026

[90] Voronin (S. M.).— Theorem on the ‘universality’ of the Riemann zeta-function, Izv. Akad. Nauk. SSSR, Ser. Matem. 39 (1975), p. 475-486 (Russian); Math. USSR Izv. 9, p. 443-445 (1975). | MR 472727 | Zbl 0315.10037

[91] Voronin (S. M.).— On the differential independence of ζ-functions, Dokl. AN SSSR 209 (6), p. 1264-1266 (Russian) (1973). | MR 319914 | Zbl 0292.10030

[92] Voronin (S. M.).— On the functional independence of Dirichlet L-functions, Acta Arithm. 27, p. 493-503 (Russian) (1975). | MR 366836 | Zbl 0308.10025

[93] Zeta function universality, Wilkipedia, 2013.