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

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.

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.

Published online:
DOI: 10.5802/afst.1419
@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 = {https://afst.centre-mersenne.org/articles/10.5802/afst.1419/}
}
TY  - JOUR
TI  - Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2014
DA  - 2014///
SP  - 621
EP  - 664
VL  - Ser. 6, 23
IS  - 3
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1419/
UR  - https://zbmath.org/?q=an%3A06374883
UR  - https://www.ams.org/mathscinet-getitem?mr=3266708
UR  - https://doi.org/10.5802/afst.1419
DO  - 10.5802/afst.1419
LA  - en
ID  - AFST_2014_6_23_3_621_0
ER  - 
%0 Journal Article
%T Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2014
%P 621-664
%V Ser. 6, 23
%N 3
%I Université Paul Sabatier, Toulouse
%U https://doi.org/10.5802/afst.1419
%R 10.5802/afst.1419
%G en
%F 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, Serie 6, Volume 23 (2014) no. 3, pp. 621-664. doi : 10.5802/afst.1419. https://afst.centre-mersenne.org/articles/10.5802/afst.1419/

[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). | EuDML: 173236 | 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). | EuDML: 158911 | 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). | EuDML: 149425 | 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.

Cited by Sources: