Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Hafedh Herichi; Michel L. Lapidus

Annales de la faculté des sciences de Toulouse Mathématiques (2014)

  • Volume: 23, Issue: 3, page 621-664
  • ISSN: 0240-2963

Abstract

top
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.

How to cite

top

Herichi, Hafedh, and 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 23.3 (2014): 621-664. <http://eudml.org/doc/275320>.

@article{Herichi2014,
abstract = {We survey some of the universality properties of the Riemann zeta function $\zeta (s)$ 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 $\{\mathfrak\{a\}\}$ (mapping the geometry onto the spectrum of generalized fractal strings), and the associated infinitesimal shift $\partial $ of the real line: $\{\mathfrak\{a\}\}=\zeta (\partial )$, in the sense of the functional calculus. In the quantization (or operator-valued) version of the universality theorem for the Riemann zeta function $\zeta (s)$ 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.},
author = {Herichi, Hafedh, Lapidus, Michel L.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
number = {3},
pages = {621-664},
publisher = {Université Paul Sabatier, Toulouse},
title = {Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function},
url = {http://eudml.org/doc/275320},
volume = {23},
year = {2014},
}

TY - JOUR
AU - Herichi, Hafedh
AU - Lapidus, Michel L.
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
PB - Université Paul Sabatier, Toulouse
VL - 23
IS - 3
SP - 621
EP - 664
AB - We survey some of the universality properties of the Riemann zeta function $\zeta (s)$ 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 ${\mathfrak{a}}$ (mapping the geometry onto the spectrum of generalized fractal strings), and the associated infinitesimal shift $\partial $ of the real line: ${\mathfrak{a}}=\zeta (\partial )$, in the sense of the functional calculus. In the quantization (or operator-valued) version of the universality theorem for the Riemann zeta function $\zeta (s)$ 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.
LA - eng
UR - http://eudml.org/doc/275320
ER -

References

top
  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). Zbl0479.10028MR678888
  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). Zbl0056.27801MR64849
  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). Zbl0764.11055MR1128186
  5. Bohr (H.).— Zur Theorie der Riemannschen ZetaFunktion im kritischen Streifen, Acta Math. 40, p. 67-100 (1915). Zbl45.0719.01MR1555133
  6. Bohr (H.).— Über eine quasi-periodische Eigenschaft Dirichletscher Reihen mit Anwendung auf die Dirichletschen L-Funktionen, Math. Ann. 85, p. 115-122 (1922). Zbl48.0343.02MR1512052
  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). Zbl45.0718.02
  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).) Zbl1220.46002MR2759829
  9. Cohn (D. L.).— Measure Theory, Birkhäuser, Boston, 1980. Zbl0860.28001MR578344
  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.) Zbl0635.47001
  11. Edwards (H. M.).— Riemann’s Zeta Function, Academic Press, New York (1974). (Paperback and reprinted edition, Dover Publications, Mineola, 2001.) Zbl1113.11303MR1854455
  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). Zbl0713.11058MR1074538
  14. Engel (K.-J.) and Nagel (R.).— One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer, Berlin (2000). Zbl0952.47036MR1721989
  15. Falconer (K. J.).— Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Chichester, (1990). (2nd edn., 2003.) Zbl0871.28009MR2118797
  16. Folland (G. B.).— Real Analysis: Modern Techniques and Their Applications, 2nd edn., John Wiley & Sons, Boston (1999). Zbl0549.28001MR1681462
  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). Zbl1070.11035MR2075625
  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). Zbl1221.30084MR2419456
  20. Gonek (S. M.).— Analytic Properties of Zeta and L -functions, Ph. D. Thesis, University of Michigan, Ann Arbor (1979). MR2628587
  21. Good (A.).— On the distribution of the values of the Riemann zeta-function, Acta Arith. 38, p. 347-388 (1981). Zbl0372.10029MR621007
  22. Haase (M.).— The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, vol. 169, Birkhäuser Verlag, Berlin (2006). Zbl1101.47010MR2244037
  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). Zbl1079.60019MR2171234
  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). Zbl0877.35086MR1376743
  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.) Zbl1252.81067MR2970522
  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.) Zbl1321.11082MR3203399
  28. Herichi (H.) and Lapidus (M. L.).— Quantized Riemann zeta function: Its operator-valued Dirichlet series, Euler product and analytic continuation, in preparation (2014). Zbl06374883
  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). Zbl0078.10004MR89373
  30. Ingham (A. E.).— The Distribution of Prime Numbers, 2nd edn. (reprinted from the 1932 edn.), Cambridge Univ. Press, Cambridge (1992). Zbl0715.11045MR1074573
  31. Ivic (A.).— The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function with Applications, John Wiley & Sons, New York (1985). Zbl0556.10026MR792089
  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).) Zbl1027.46002MR1771173
  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). Zbl0139.05603MR201237
  34. Karatsuba (A. A.) and Voronin (S. M.).— The Riemann Zeta-Function, Expositions in Mathematics, Walter de Gruyter, Berlin (1992). Zbl0756.11022MR1183467
  35. Kato (T.).— Perturbation Theory for Linear Operators, Springer-Verlag, New York (1995). Zbl0435.47001MR1335452
  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.) Zbl1256.28004MR2967908
  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.) MR3203865
  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). Zbl0741.35048MR994168
  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). Zbl0736.58040MR1126694
  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). Zbl0830.35094MR1234502
  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.) Zbl0870.58063MR1410291
  42. Lapidus (M. L.).— In Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes, Amer. Math. Soc., Providence, R. I. (2008). Zbl1150.11003MR2375028
  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.) Zbl1170.11030MR2512142
  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.) Zbl1187.28014MR2566062
  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.) Zbl1276.37053MR2882397
  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.) Zbl1276.00022MR3203402
  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). Zbl0751.35030MR1115940
  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). Zbl0836.11031MR1345711
  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.) Zbl1244.28013MR2684976
  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.) Zbl1274.28016MR2799798
  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). Zbl0707.58046MR1046509
  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). Zbl0739.34065MR1189091
  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). Zbl0858.58052MR1356166
  56. Lapidus (M. L.), Radunović (G.) and Z ˇ ubrinić (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). Zbl0945.11016MR1710790
  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). Zbl0981.28005MR1726744
  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). Zbl1119.28005MR2245559
  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). Zbl1261.28011MR2245559
  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). Zbl0845.11002MR1376140
  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). Zbl0938.11045MR1481388
  64. Laurincikas (A.).— Prehistory of the Voronin universality theorem, Siauliai Math. J. 1 (9), p. 41-53 (2006). Zbl1126.11041MR2547354
  65. Laurincikas (A.) and Matsumoto (K.).— The joint universality of twisted automorphic L-functions, J. Math. Soc. Japan 56, p. 923-939 (2004). Zbl1142.11032MR2071679
  66. Laurincikas (A.) and Matsumoto (K.).— The universality of zeta-functions attached to certain cusp forms, Acta Arith. 98, p. 345-359 (2001). Zbl0974.11018MR1829777
  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). Zbl1112.11026MR1957917
  68. Laurincikas (A.) and Slezeviciene (R.).— The universality of zeta-functions with multiplicative coefficients, Integral Transforms and Special Functions 13, p. 243-257 (2002). Zbl1020.11057MR1919181
  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). Zbl1155.11033MR2310179
  70. Mandelbrot (B. B.).— The Fractal Geometry of Nature, rev. and enl. edn. (of the 1977 edn.), W. H. Freeman, New York (1983). Zbl0504.28001MR665254
  71. Mattila (P.).— Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge Univ. Press, Cambridge (1995). Zbl0911.28005MR1333890
  72. Montgomery (H. L.).— Extreme values of the Riemann zeta-function, Comment. Math. Helv. 52, p. 511-518 (1997). Zbl0373.10024MR460255
  73. Ostrowski (A.).— Über Dirichletsche Reihen und algebraische Differentialgleichungen, Math. Z. 8, 241-298 (1920). Zbl47.0292.01MR1544442
  74. Patterson (S. J.).— An Introduction to the Theory of the Riemann Zeta-Function, Cambridge Univ. Press, Cambridge (1988). Zbl0831.11045MR933558
  75. Pazy (A.).— Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin and New York (1983). Zbl0516.47023MR710486
  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.) Zbl1217.41023MR2587018
  77. Reich (A.).— Wertverteilung von Zetafunktionen, Arch. Math. 34, p. 440-451 (1980). Zbl0431.10025MR593771
  78. Reich (A.).— Universelle Wertevereteilung von Eulerprodukten, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, Nos. 1-17 (1977). Zbl0379.10025MR567687
  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. Zbl0308.47002MR751959
  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). Zbl0867.46001MR1157815
  82. Schechter (M.).— Operator Methods in Quantum Mechanics, Dover Publications, New York (2003). Zbl1029.47054MR1969612
  83. Schwartz (L.).— Théorie des Distributions, rev. and enl. edn. (of the 1951 edn.), Hermann, Paris (1996). Zbl0042.11405MR41345
  84. Serre (J.-P.).— A Course in Arithmetic, English transl., Springer-Verlag, Berlin (1973). Zbl0432.10001MR344216
  85. Steuding (J.).— Value-Distribution of L -Functions, Lecture Notes in Mathematics, vol. 1877, Springer, Berlin (2007). Zbl1130.11044MR2330696
  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). Zbl1059.11052MR2075637
  87. Teplyaev (A.).— Spectral zeta functions of fractals and the complex dynamics of polynomials, Trans. Amer. Math. Soc. 359, p. 4339-4358 (2007). Zbl1129.28010MR2309188
  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). Zbl0601.10026MR882550
  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). Zbl0294.10026MR319915
  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). Zbl0315.10037MR472727
  91. Voronin (S. M.).— On the differential independence of ζ -functions, Dokl. AN SSSR 209 (6), p. 1264-1266 (Russian) (1973). Zbl0292.10030MR319914
  92. Voronin (S. M.).— On the functional independence of Dirichlet L -functions, Acta Arithm. 27, p. 493-503 (Russian) (1975). Zbl0308.10025MR366836
  93. Zeta function universality, Wilkipedia, 2013. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.