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The Sound of Fractal Strings and the Riemann Hypothesis

  • Chapter
Analytic Number Theory

Abstract

We give an overview of the intimate connections between natural direct and inverse spectral problems for fractal strings, on the one hand, and the Riemann zeta function and the Riemann hypothesis, on the other hand (in joint works of the author with Carl Pomerance and Helmut Maier, respectively). We also briefly discuss closely related developments, including the theory of (fractal) complex dimensions (by the author and many of his collaborators, including especially Machiel van Frankenhuijsen), quantized number theory and the spectral operator (jointly with Hafedh Herichi), and some other works of the author (and several of his collaborators).

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Notes

  1. 1.

    This is really the upper Minkowski dimension of ∂ Ω, relative to Ω, but we will not stress this point in this paper (except perhaps in Sect. 7.3).

  2. 2.

    A simple counterexample is provided by \(A:=\{ 1/k: k \geq 1\}\) and \(A_{k}:=\{ \frac{1} {k}\}\) for each k ≥ 1, viewed as subsets of \(\mathbb{R}\); then, \(D(A) = 1/2,\) whereas D(A k ) = 0 for all k ≥ 1 and hence, \(\sup _{k\geq 1}D(A_{k}) = 0.\)

  3. 3.

    See, e.g., [55–57, 60–62, 71, 124, 125, 162–164] (along with the relevant references therein and in [101, §12.5 and Appendix B]) for results (in the smooth case) concerning the Weyl conjecture about the asymptotic second term of the spectral counting function N ν (x).

  4. 4.

    Strictly speaking, when \(D = N - 1,N - D = 1\) is not equal to {1}.

  5. 5.

    It can now also be systematically understood in terms of the generalized explicit formulas of [101, Chap. 5]; see [101, Chap. 9]. The resulting theorems and assumptions, however, are somewhat different.

  6. 6.

    This is now proved in [99–101] by using the (generalized) explicit formulas from [99–101] and Eq. (1).

  7. 7.

    We note that the product formula (1) for the spectral zeta functions of fractal strings has since been extended in various ways in the setting of Laplacians on certain self-similar fractals; see [26, 72, 73, 172, 173].

  8. 8.

    An example of a trivial self-similar set is an interval of \(\mathbb{R}\) or a cube in \(\mathbb{R}^{N}(N \geq 2).\) In such cases, all of the complex dimensions are easily seen to be real; see [101, 112, 113] and [116].

  9. 9.

    Since ζ A and \(\tilde{\zeta }_{A}\) are holomorphic in the open half-plane {Re(s) > D}, it then follows that

    $$\displaystyle{ D =\max \{ \text{Re}(s): s \in \mathcal{D}\}, }$$
    (42)

    where \(\mathcal{D}\) is the set of (visible) complex dimensions of A in any domain U containing this neighborhood of {D}.

  10. 10.

    Recall that in the terminology of [108–113, 115, 116], a “maximal hyperfractal” is a compact subset A of \(\mathbb{R}^{N}\) such that ζ A has a nonremovable singularity at every point of the “critical line” {Re(s) = D}; in addition to Remark 7.19, see Remark 7.22 below.

  11. 11.

    Added note: In the case of self-similar sprays, the results of [108, 112, 113, 116] now enable one to recover and significantly extend the results of [91, 92, 104, 105].

  12. 12.

    Hence, \(\mathbb{H}_{c}\) is the complex Hilbert space of (Lebesgue measurable, complex-valued) square integrable functions with respect to the absolutely continuous measure e −2ct dt, and is equipped with the norm

    $$\displaystyle{ \vert \vert f\vert \vert _{c}:= \left (\int _{\mathbb{R}}\vert f(t)\vert ^{2}\ e^{-2ct}\ dt\right )^{1/2} <\infty }$$
    (55)

    and the associated inner product, denoted by (⋅ , ⋅ ) c .

  13. 13.

    In fact, ζ has infinitely many zeros along the critical line, according to Hardy’s theorem (see [29, 174]) but this is irrelevant here. We refer to [50, 53] for a version of Theorem 7.26 for which this well-known fact actually matters.

  14. 14.

    A similar phase transition (or “symmetry breaking”) is observed at c = 1, as is discussed in [49–51, 53], since \(\mathfrak{a}\) is bounded and invertible for c > 1, unbounded and not invertible for 1∕2 ≤ c ≤ 1, while, likewise, \(\sigma (\mathfrak{a})\) is a compact subset of \(\mathbb{C}\) not containing the origin for c > 1 and \(\sigma (\mathfrak{a}) = \mathbb{C}\) is unbounded and contains the origin if 1∕2 < c < 1.

  15. 15.

    See, e.g., [79, Appendix B] and the references therein.

  16. 16.

    See, e.g., [79, Chaps. 4 and 5], along with the relevant references therein.

  17. 17.

    Some of this work may eventually become joint work with one of the author’s current Ph.D. students, Tim Cobler.

References

  1. A. Atzmon, B. Brive, Surjectivity and invariant subspaces of differential operators on weighted Bergman spaces of entire functions, in Bergman Spaces and Related Topics in Complex Analysis, ed. by A. Borichev, H. Hedenmalm, K. Zhu. Contemporary Mathematics, vol. 404 (American Mathematical Society, Providence, RI, 2006), pp. 27–39

    Google Scholar 

  2. B. Bagchi, The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series. Ph.D. thesis, Indian Statistical Institute, Calcutta, 1981

    Google Scholar 

  3. B. Bagchi, A joint universality theorem for Dirichlet L-functions. Math. Z. 181, 319–334 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Baker, Transcendental Number Theory (Cambridge University Press, Cambridge, 1975)

    Book  MATH  Google Scholar 

  5. T. Bedford, M. Keane, C. Series (eds.), Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces (Oxford University Press, Oxford, 1991)

    MATH  Google Scholar 

  6. M.V. Berry, Distribution of modes in fractal resonators, in Structural Stability in Physics, ed. by W. Güttinger, H. Eikemeier. Graduate Texts in Mathematics, vol. 125 (Springer, Berlin, 1979), pp. 51–53

    Google Scholar 

  7. M.V. Berry, Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals, in Geometry of the Laplace Operator. Proceedings of Symposia in Pure Mathematics, vol. 36 (American Mathematical Society, Providence, RI, 1980), pp. 13–38

    Google Scholar 

  8. A.S. Besicovitch, S.J. Taylor, On the complementary intervals of a linear closed set of zero Lebesgue measure. J. Lond. Math. Soc. 29, 449–459 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Bohr, Zur Theorie der Riemannschen ZetaFunktion im kritischen Streifen. Acta Math. 40, 67–100 (1915)

    Article  MathSciNet  MATH  Google Scholar 

  10. H. Bohr, Über eine quasi-periodische Eigenschaft Dirichletscher Reihen mit Anwendung auf die Dirichletschen L-Funktionen. Math. Ann. 85, 115–122 (1922)

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Bohr, R. Courant, Neue Anwendungen der Theorie der diophantischen Approximationen auf die Riemannsche Zetafunktion. J. Reine Angew. Math. 144, 249–274 (1914)

    MathSciNet  MATH  Google Scholar 

  12. H. Bohr, E. Landau, Über das Verhalten von ζ(s) und ζ (k)(s) in der Nähe der Geraden \(\sigma = 1\). Nachr. Ges. Wiss. Göttingen Math. Phys. K1, 303–330 (1910)

    Google Scholar 

  13. H. Bohr, E. Landau, Ein Satz über Dirichletsche Reihen mit Anwendung auf die ζ-Funktion und die L-Funktionen. Rend. di Palermo 37, 269–272 (1914)

    Article  MATH  Google Scholar 

  14. J.-B. Bost, A. Connes, Produit eulérien et facteurs de type III. C. R. Acad. Sci. Paris Sér. I Math. 315, 279–284 (1992)

    MathSciNet  MATH  Google Scholar 

  15. J.-B. Bost, A. Connes, Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory. Sel. Math. N.S. 1, 411–457 (1995)

    Google Scholar 

  16. G. Bouligand, Ensembles impropres et nombre dimensionnel. Bull. Sci. Math. 52(2), 320–344/361–376 (1928)

    Google Scholar 

  17. H. Brezis, Analyse Fonctionnelle: Théorie et Applications (Masson, Paris, 1983). Expanded English version: Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer, New York, 2011)

    Google Scholar 

  18. J. Brossard, R. Carmona, Can one hear the dimension of a fractal? Commun. Math. Phys. 104, 103–122 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Cartier, A mad day’s work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry (English transl. of the French original). Bull. Am. Math. Soc. (N.S.) 38, 389–408 (2001)

    Google Scholar 

  20. E. Christensen, C. Ivan, M.L. Lapidus, Dirac operators and spectral triples for some fractal sets built on curves. Adv. Math. 217(1), 42–78 (2008). Also: e-print, arXiv:math.MG/0610222v2, 2007

    Google Scholar 

  21. D.L. Cohn, Measure Theory (Birkhäuser, Boston, 1980)

    Book  MATH  Google Scholar 

  22. A. Connes, Noncommutative Geometry (Academic, New York, 1994)

    MATH  Google Scholar 

  23. R. Courant, D. Hilbert, Methods of Mathematical Physics, vol. I (Interscience Publishers Inc., New York, 1953)

    Google Scholar 

  24. C. Deninger, Lefschetz trace formulas and explicit formulas in analytic number theory. J. Reine Angew. Math. 441, 1–15 (1993)

    MathSciNet  MATH  Google Scholar 

  25. C. Deninger, Evidence for a cohomological approach to analytic number theory, in Proceedings of First European Congress of Mathematics, Paris, July 1992, vol. I, ed. by A. Joseph et al. (Birkhäuser-Verlag, Basel and Boston, 1994), pp. 491–510

    Chapter  Google Scholar 

  26. G. Derfel, P. Grabner, F. Vogl, The zeta function of the Laplacian on certain fractals. Trans. Amer. Math. Soc. 360, 881–897 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. J.D. Dollard, C.N. Friedman, Product Integration, with Application to Differential Equations. Encyclopedia of Mathematics and Its Applications, vol. 10 (Addison-Wesley, Reading, 1979)

    Google Scholar 

  28. N. Dunford, J.T. Schwartz, Linear Operators. Wiley Classics Library, Parts I–III (Wiley, Hoboken, 1971/1988)

    Google Scholar 

  29. H.M. Edwards, Riemann’s Zeta Function (Academic, New York, 1974)

    MATH  Google Scholar 

  30. K.E. Ellis, M.L. Lapidus, M.C. Mackenzie, J.A. Rock, Partition zeta functions, multifractal spectra, and tapestries of complex dimensions, in Benoît Mandelbrot: A Life in Many Dimensions, ed. by M. Frame, N. Cohen. The Mandelbrot Memorial Volume (World Scientific, Singapore, 2015), pp. 267–322. Also: e-print, arXiv:1007.1467v2[math-ph], 2011; IHES preprint, IHES/M/12/15, 2012

    Google Scholar 

  31. K.J. Falconer, On the Minkowski measurability of fractals. Proc. Amer. Math. Soc. 123, 1115–1124 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. K.J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3rd edn. (Wiley, Chichester, 2014). First and second editions: 1990 and 2003

    Google Scholar 

  33. H. Federer, Geometric Measure Theory (Springer, New York, 1969)

    MATH  Google Scholar 

  34. J. Fleckinger, D. Vassiliev, An example of a two-term asymptotics for the “counting function” of a fractal drum. Trans. Amer. Math. Soc. 337, 99–116 (1993)

    MathSciNet  MATH  Google Scholar 

  35. G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edn. (Wiley, New York, 1999)

    MATH  Google Scholar 

  36. M. Fukushima, T. Shima, On a spectral analysis for the Sierpinski gasket. Potential Anal. 1, 1–35 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. R. Garunks̆tis, J. Steuding, On the roots of the equation ζ(s) = α. Abh. Math. Seminar Univ. Hamburg 84, 1–15 (2014). Also: e-print, arXiv:1011.5339v2 [math.NT], 2014

    Google Scholar 

  38. J. Gerling, Untersuchungen zur Theorie von Weyl–Berry–Lapidus. Graduate thesis (Diplomarbeit), Department of Physics, Universität Osnabrück, Osnabrück, 1992

    Google Scholar 

  39. J. Gerling, H.-J. Schmidt, Self-similar drums and generalized Weierstrass functions. Physica A 191(1–4), 536–539 (1992)

    Article  MathSciNet  Google Scholar 

  40. P.B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2nd edn. (Publish or Perish, Wilmington, 1984). New revised and enlarged edition in Studies in Advanced Mathematics (CRC Press, Boca Raton, 1995)

    Google Scholar 

  41. J.A. Goldstein, Semigroups of Linear Operators and Applications. Oxford Science Publications, Oxford Mathematical Monographs (Oxford University Press, Oxford and New York, 1985)

    MATH  Google Scholar 

  42. B. M. Hambly, Brownian motion on a random recursive Sierpinski gasket. Ann. Probab. 25, 1059–1102 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. B.M. Hambly, On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets. Probab. Theory Relat. Fields 117, 221–247 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  44. B.M. Hambly, M.L. Lapidus, Random fractal strings: their zeta functions, complex dimensions and spectral asymptotics. Trans. Amer. Math. Soc. 358(1), 285–314 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 6th edn. (Oxford University Press, Oxford, 2008)

    MATH  Google Scholar 

  46. R. Harvey, J. Polking, Removable singularities of solutions of linear partial differential equations. Acta Math. 125, 39–56 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  47. C.Q. He, M.L. Lapidus, Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta-function. Mem. Amer. Math. Soc. 127(608), 1–97 (1997)

    MathSciNet  Google Scholar 

  48. H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces. Graduate Texts in Mathematics, vol. 199 (Springer, New York, 2000)

    Google Scholar 

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

    Google Scholar 

  50. H. Herichi, M.L. Lapidus, 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, ed. by D. Carfi, M.L. Lapidus, E.P.J. Pearse, M. van Frankenhuijsen. Contemporary Mathematics, vol. 600 (American Mathematical Society, Providence, RI, 2013), pp. 51–89. Also: e-print, arXiv:1210.0882v3[math.FA], 2013; IHES preprint, IHES/M/12/25, 2012

    Google Scholar 

  51. H. Herichi, M.L. Lapidus, Truncated infinitesimal shifts, spectral operators and quantized universality of the Riemann zeta function. Annales de la Faculté des Sciences de Toulouse 23(3), 621–664 (2014). Special issue in honor of Christophe Soulé. Also: e-print, arXiv:1305.3933v2[math-NT], 2015; IHES preprint, IHES/M/13/12, 2013

    Google Scholar 

  52. H. Herichi, M.L. Lapidus, Quantized Riemann zeta functions: its operator-valued Dirichlet series, Euler product and analytic continuation (2015, in preparation)

    Google Scholar 

  53. H. Herichi, M. L. Lapidus, Quantized Number Theory, Fractal Strings and the Riemann Hypothesis: From Spectral Operators to Phase Transitions and Universality. Research Monograph (World Scientific Publ., Singapore, 2016, to appear). Approx. 240 pp.

    Google Scholar 

  54. E. Hille, R.S. Phillips, Functional Analysis and Semi-Groups. American Mathematical Society Colloquium Publications, vol. XXXI, revised edn. (American Mathematical Society, RI, 1957)

    Google Scholar 

  55. L. Hörmander, The spectral function of an elliptic operator. Acta Math. 121, 193–218 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  56. L. Hörmander, The Analysis of Linear Partial Differential Operators, vols. II–IV (Springer, Berlin, 1983/1985)

    Google Scholar 

  57. L. Hörmander, The Analysis of Linear Partial Differential Operators. Distribution Theory and Fourier Analysis, vol. I, 2nd edn. (of the 1983 edn.) (Springer, Berlin, 1990)

    Google Scholar 

  58. A.E. Ingham, The Distribution of Prime Numbers, 2nd edn. (reprinted from the 1932 edn.) (Cambridge University Press, Cambridge, 1992)

    Google Scholar 

  59. A. Ivic, The Riemann Zeta-Function: The Theory of the Riemann Zeta-Function with Applications (Wiley, New York, 1985)

    MATH  Google Scholar 

  60. V.Ja. Ivrii, Second term of the spectral asymptotic expansion of the Laplace-Beltrami operator on manifolds with boundary. Funct. Anal. Appl. 14, 98–106 (1980)

    Article  Google Scholar 

  61. V.Ja. Ivrii, Precise Spectral Asymptotics for Elliptic Operators Acting in Fiberings over Manifolds with Boundary. Lecture Notes in Mathematics, vol. 1100 (Springer, New York, 1984)

    Google Scholar 

  62. V.Ja. Ivrii, Microlocal Analysis and Precise Spectral Asymptotics (Springer, Berlin, 1998)

    Google Scholar 

  63. G.W. Johnson, M.L. Lapidus, The Feynman Integral and Feynman’s Operational Calculus. Oxford Science Publications, Oxford Mathematical Monographs (Oxford University Press, Oxford and New York, 2000). Corrected printing and paperback edition, 2002

    Google Scholar 

  64. G.W. Johnson, M.L. Lapidus, L. Nielsen, Feynman’s Operational Calculus and Beyond: Noncommutativity and Time-Ordering. Oxford Science Publications, Oxford Mathematical Monographs (Oxford University Press, Oxford and New York, 2015); ISBN 978-0-19-870249-8. Approx. 400 pp.

    Google Scholar 

  65. M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly (Slaught Memorial Papers, No. 11) 73(4), 1–23 (1966)

    Google Scholar 

  66. A.A. Karatsuba, S.M. Voronin, The Riemann Zeta-Function. De Gruyter, Expositions in Mathematics (Walter de Gruyter, Berlin, 1992)

    Google Scholar 

  67. T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1995)

    MATH  Google Scholar 

  68. J. Kigami, Analysis on Fractals (Cambridge University Press, Cambridge, 2001)

    Book  MATH  Google Scholar 

  69. J. Kigami, M.L. Lapidus, Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals. Commun. Math. Phys. 158, 93–125 (1993)

    Google Scholar 

  70. J. Kigami, M.L. Lapidus, Self-similarity of volume measures for Laplacians on p.c.f. self-similar fractals. Commun. Math. Phys. 217, 165–180 (2001)

    Google Scholar 

  71. P.T. Lai, Meilleures estimations asymptotiques des restes de la fonction spectrale et des valeurs propres relatifs au laplacien. Math. Scand. 48, 5–38 (1981)

    MathSciNet  MATH  Google Scholar 

  72. N. Lal, M.L. Lapidus, Hyperfunctions and spectral zeta functions of Laplacians on self-similar fractals. J. Phys. A Math. Theor. 45, 365205, l4 pp. (2012). Also: e-print, arXiv:12O2.4126v2[math-ph], 2012; IHES preprint, IHES/M/12/14, 2012

    Google Scholar 

  73. N. Lal, M.L. Lapidus, 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, ed. by D. Carfi, M.L. Lapidus, E.P.J. Pearse, M. van Frankenhuijsen. Contemporary Mathematics, vol. 601 (American Mathematical Society, Providence, RI, 2013), pp. 227–249. Also: e-print, arXiv:1302.4007v2[math-ph], 2014; IHES preprint, IHES/M/12/31, 2012

    Google Scholar 

  74. M.L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl–Berry conjecture. Trans. Amer. Math. Soc. 325, 465–529 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  75. M.L. Lapidus, 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, ed. by C. Bennewitz. Proceedings of Fourth UAB International Conference, Birmingham, March 1990 (Academic, New York, 1992), pp. 151–182

    Google Scholar 

  76. M.L. Lapidus, Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl–Berry conjecture, in Ordinary and Partial Differential Equations, ed. by B.D. Sleeman, R.J. Jarvis. Vol. IV, Proceedings of Twelfth International Conference (Dundee, Scotland, UK, June 1992), Pitman Research Notes in Math. Series, vol. 289 (Longman Scientific and Technical, London, 1993), pp. 126–209

    Google Scholar 

  77. M.L. Lapidus, Analysis on fractals, Laplacians on self-similar sets, noncommutative geometry and spectral dimensions. Topol. Methods Nonlinear Anal. 4, 137–195 (1994). Special issue dedicated to Jean Leray

    Google Scholar 

  78. M.L. Lapidus, Towards a noncommutative fractal geometry? Laplacians and volume measures on fractals, in Harmonic Analysis and Nonlinear Differential Equations: A Volume in Honor of Victor L. Shapiro. Contemporary Mathematics, vol. 208 (American Mathematical Society, Providence, RI, 1997), pp. 211–252

    Google Scholar 

  79. M.L. Lapidus, In Search of the Riemann Zeros: Strings, Fractal Membranes and Noncommutative Spacetimes (American Mathematical Society, Providence, RI, 2008)

    Google Scholar 

  80. M.L. Lapidus, Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis. Philos. Trans. Royal Soc. Ser. A No. 2047, 373, 24 pp. (2015); doi:10.1098/rsta.2014.0240. Special issue titled “Geometric concepts in the foundations of physics”. (Also: e-print, arXiv:1501.05362v2 [math-ph], 2015; IHES preprint, IHES/M/15/12, 2015.)

  81. M.L. Lapidus, Riemann hypothesis, weighted Bergman spaces and quantized Riemann zeta function (tentative title) (2015, in preparation)

    Google Scholar 

  82. M.L. Lapidus, Quantized Weil conjectures, spectral operators and Pólya–Hilbert operators (tentative title) (2015, in preparation)

    Google Scholar 

  83. M.L. Lapidus, H. Lu, Nonarchimedean Cantor set and string. J. Fixed Point Theory Appl. 3, 181–190 (2008). Special issue dedicated to the Jubilee of Vladimir I. Arnold, vol. I

    Google Scholar 

  84. M.L. Lapidus, H. Lu, Self-similar p-adic fractal strings and their complex dimensions. p-adic Numbers Ultrametric Anal. Appl. (Springer & Russian Academy of Sciences, Moscow) 1(2), 167–180 (2009). Also: IHES preprint, IHES/M/08/42, 2008

    Google Scholar 

  85. M.L. Lapidus, H. Lu, The geometry of p-adic fractal strings: a comparative survey, in Advances in Non-Archimedean Analysis, ed. by J. Araujo, B. Diarra, A. Escassut. Proceedings of 11th International Conference on p-Adic Functional Analysis (Clermont-Ferrand, France, July 2010). Contemporary Mathematics, vol. 551 (American Mathematical Society, Providence, RI, 2011), pp. 163–206. Also: e-print, arXiv:1105.2966v1 [math.MG], 2011

    Google Scholar 

  86. M. L. Lapidus, H. Maier, Hypothèse de Riemann, cordes fractales vibrantes et conjecture de Weyl–Berry modifiée. C. R. Acad. Sci. Paris Sér. I Math. 313, 19–24 (1991)

    MathSciNet  MATH  Google Scholar 

  87. M.L. Lapidus, H. Maier, The Riemann hypothesis and inverse spectral problems for fractal strings. J. Lond. Math. Soc. 52(2), 15–34 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  88. M.L. Lapidus, R. Nest, Fractal membranes as the second quantization of fractal strings. (preliminary) (2015, preprint)

    Google Scholar 

  89. M.L. Lapidus, M.M.H. Pang, Eigenfunctions of the Koch snowflake drum. Commun. Math. Phys. 172, 359–376 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  90. M.L. Lapidus, E.P.J. Pearse, A tube formula for the Koch snowflake curve, with applications to complex dimensions. J. Lond. Math. Soc. 74(2), 397–414 (2006). Also: e-print, arXiv:math-ph/0412029v2, 2005

    Google Scholar 

  91. M.L. Lapidus, E.P.J. Pearse, Tube formulas for self-similar fractals, in Analysis on Graphs and Its Applications, ed. by P. Exner et al. Proceedings of Symposia in Pure Mathematics, vol. 77 (American Mathematical Society, Providence, RI, 2008), pp. 211–230. Also: e-print, arXiv:math.DS/0711.0173, 2007; IHES preprint, IHES/M/08/28, 2008

    Google Scholar 

  92. M.L. Lapidus, E.P.J. Pearse, Tube formulas and complex dimensions of self-similar tilings. Acta Appl. Math. 112(1), 91–137 (2010). Springer Open Access: doi:10.1007/S10440-010-9562-x. Also: e-print, arXiv:math.DS/0605527v5, 2010; IHES preprint, IHES/M/08/27, 2008

  93. M.L. Lapidus, C. Pomerance, Fonction zêta de Riemann et conjecture de Weyl–Berry pour les tambours fractals. C. R. Acad. Sci. Paris Sér. I Math. 310, 343–348 (1990)

    MathSciNet  MATH  Google Scholar 

  94. M.L. Lapidus, C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl–Berry conjecture for fractal drums. Proc. Lond. Math. Soc. 66(1), 41–69 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  95. M.L. Lapidus, C. Pomerance, Counterexamples to the modified Weyl–Berry conjecture on fractal drums. Math. Proc. Camb. Philos. Soc. 119, 167–178 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  96. M.L. Lapidus, J.A. Rock, Towards zeta functions and complex dimensions of multifractals. Complex Variables Elliptic Equ. 54(6), 545–560 (2009). Also: e-print, arXiv:math-ph/0810.0789, 2008

    Google Scholar 

  97. M.L. Lapidus, J.A. Rock, An Invitation to Fractal Geometry: Dimension Theory, Zeta Functions and Applications (2015, in preparation)

    Google Scholar 

  98. M.L. Lapidus, J.J. Sarhad, Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets. J. Noncommutative Geometry 8(4), 947–985 (2014). doi:10.4171/JNCG/174. Also: e-print, arXiv:1212:0878v2[math.MG], 2014; IHES preprint, IHES/M/12/32, 2012

  99. M.L. Lapidus, M. van Frankenhuijsen, Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions (Birkhäuser, Boston, 2000)

    Book  Google Scholar 

  100. M.L. Lapidus, M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings. Springer Monographs in Mathematics (Springer, New York, 2006)

    Google Scholar 

  101. M.L. Lapidus, M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2nd revised and enlarged edition (of the 2006 edn., [100]). Springer Monographs in Mathematics (Springer, New York, 2013)

    Google Scholar 

  102. M.L. Lapidus, J.W. Neuberger, R.J. Renka, C.A. Griffith, Snowflake harmonics and computer graphics: numerical computation of spectra on fractal domains. Int. J. Bifurcation Chaos 6, 1185–1210 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  103. M.L. Lapidus, J. Lévy-Véhel, J.A. Rock, Fractal strings and multifractal zeta functions. Lett. Math. Phys. 88(1), 101–129 (2009). Springer Open Access, doi:10.1007/s1105-009-0302-y. Also: e-print, arXiv:math-ph/0610015v3, 2009

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

    Google Scholar 

  105. M.L. Lapidus, E.P.J. Pearse, S. Winter, Minkowski measurability results for self-similar tilings and fractals with monophase generators, in Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I: Fractals in Pure Mathematics, ed. by D. Carfi, M.L. Lapidus, E.P.J. Pearse, M. van Frankenhuijsen. Contemporary Mathematics, vol. 600 (American Mathematical Society, Providence, RI, 2013), pp. 185–203. Also: e-print, arXiv:1104.1641v3[math.MG], 2012; IHES preprint, IHES/M/12/33, 2012

    Google Scholar 

  106. M.L. Lapidus, J.A. Rock, D. Z̆ubrinić, Box-counting fractal strings, zeta functions, and equivalent forms of Minkowski dimension, in Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I: Fractals in Pure Mathematics, ed. by D. Carfi, M.L. Lapidus, E.P.J. Pearse, M. van Frankenhuijsen. Contemporary Mathematics, vol. 600 (American Mathematical Society, Providence, RI, 2013), pp. 239–271. Also: e-print, arXiv:1207.6681v2 [math-ph], 2013; IHES preprint, IHES/M/12/22, 2012

    Google Scholar 

  107. M.L. Lapidus, H. Lu, M. van Frankenhuijsen, 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, ed. by D. Carfi, M.L. Lapidus, E.P. J. Pearse, M. van Frankenhuijsen. Contemporary Mathematics, vol. 600 (American Mathematical Society, Providence, RI, 2013), pp. 185–203. Also: e-print, arXiv:1209.6440v1[math.MG], 2012; IHES preprint, IHES/M/12/23, 2012

    Google Scholar 

  108. M.L. Lapidus, G. Radunović, D. Z̆ubrinić, Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces (2015). Also: e-print, arXiv:1411.5733v2[math-ph], 2015; IHES preprint, IHES/M/15/17, 2015

    Google Scholar 

  109. M.L. Lapidus, G. Radunović, D. Z̆ubrinić, Fractal zeta functions and complex dimensions of relative fractal drums, survey article. J. Fixed Point Theory Appl. 15(2), 321–378 (2014). Festschrift issue in honor of Haim Brezis’ 70th birthday. doi:10.1007/s11784-014-0207-y. Also: e-print, arXiv:1407.8094v3[math-ph], 2014; IHES preprint, IHES/M/15/14, 2015

  110. M.L. Lapidus, G. Radunović, D. Z̆ubrinić, Distance and tube zeta functions of fractals and arbitrary compact sets (2015, preprint). Also: e-print, arXiv:1506.03525v2 [math-ph], 2015; IHES preprint, IHES/M/15/15, 2015

    Google Scholar 

  111. M.L. Lapidus, G. Radunović, D. Z̆ubrinić, Complex dimensions of fractals and meromorphic extensions of fractal zeta functions (2015, preprint). Also: e-print, arXiv:1508.04784v1 [math-ph], 2015

    Google Scholar 

  112. M.L. Lapidus, G. Radunović, D. Z̆ubrinić, Zeta functions and complex dimensions of relative fractal drums: theory, examples and applications (2015, preprint)

    Google Scholar 

  113. M.L. Lapidus, G. Radunović, D. Z̆ubrinić, Fractal tube formulas for compact sets and relative fractal drums, with application to a Minkowski measurability criterion (2015, preprint)

    Google Scholar 

  114. M.L. Lapidus, H. Lu, M. van Frankenhuijsen, Minkowski dimension and explicit tube formulas for p-adic fractal strings (2015, preprint)

    Google Scholar 

  115. M.L. Lapidus, G. Radunović, D. Z̆ubrinić, Fractal zeta functions and complex dimensions: a general higher-dimensional theory, survey article, in Geometry and Stochastics V, ed. by C. Bandt, K. Falconer, M. Zähle. Proceedings of Fifth International Conference (Tabarz, Germany, March 2014). Progress in Probability (Birkhäuser, Basel, 2015, pp. 229–257); doi:10.1007/978-3-319-18660-3_13. Based on a plenary lecture given by the first author at that conference. Also: e-print, arXiv:1502.00878v3[math.CV], 2015; IHES preprint, IHES/M/15/16, 2015

  116. M.L. Lapidus, G. Radunović, D. Z̆ubrinić, Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions. Research Monograph (Springer, New York, 2016, to appear). Approx. 625 pp.

    Google Scholar 

  117. A. Laurincikas, Limit Theorems for the Riemann Zeta-Function (Kluwer Academic Publishers, Dordrecht, 1996)

    Book  Google Scholar 

  118. T. Lei (ed.), The Mandelbrot Set, Theme and Variations. London Mathematical Society Lecture Notes Series, vol. 274 (Cambridge University Press, Cambridge, 2000)

    Google Scholar 

  119. J. Lévy-Véhel, F. Mendivil, Multifractal and higher-dimensional zeta functions. Nonlinearity 24(1), 259–276 (2011)

    Article  MathSciNet  Google Scholar 

  120. J.L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vol. I, English transl. (Springer, Berlin, 1972)

    Google Scholar 

  121. B.B. Mandelbrot, The Fractal Geometry of Nature, revised and enlarged edition (of the 1977 edn.) (W. H. Freeman, New York, 1983)

    Google Scholar 

  122. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability (Cambridge University Press, Cambridge, 1995)

    Book  MATH  Google Scholar 

  123. V.G. Maz’ja, Sobolev Spaces (Springer, Berlin, 1985)

    Book  MATH  Google Scholar 

  124. R.B. Melrose, Weyl’s conjecture for manifolds with concave boundary, in Geometry of the Laplace Operator. Proceedings of Symposia in Pure Mathematics, vol. 36 (American Mathematical Society, Providence, RI, 1980), pp. 254–274

    Google Scholar 

  125. R.B. Melrose, The Trace of the Wave Group. Contemporary Mathematics, vol. 27 (American Mathematical Society, Providence, RI, 1984), pp. 127–167

    Google Scholar 

  126. G. Métivier, Théorie spectrale d’opérateurs elliptiques sur des ouverts irréguliers, Séminaire Goulaic-Schwartz, No. 21 (Ecole Polytechnique, Paris, 1973)

    Google Scholar 

  127. G. Métivier, Etude asymptotique des valeurs propres et de la fonction spectrale de problèmes aux limites. Thèse de Doctorat d’Etat, Mathématiques, Université de Nice, 1976

    Google Scholar 

  128. G. Métivier, Valeurs propres de problèmes aux limites elliptiques irréguliers. Bull. Soc. Math. France Mém. 51–52, 125–219 (1977)

    Google Scholar 

  129. S. Molchanov, B. Vainberg, On spectral asymptotics for domains with fractal boundaries. Commun. Math. Phys. 183, 85–117 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  130. G. Mora, J.M. Sepulcre, T. Vidal, On the existence of exponential polynomials with prefixed gaps. Bull. Lond. Math. Soc. 45(6), 1148–1162 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  131. A.N. Parshin, I.R. Shafarevich (eds.), Number Theory, vol. II, Algebraic Number Fields. Encyclopedia of Mathematical Sciences, vol. 62 (Springer, Berlin, 1992). Written by H. Koch.

    Google Scholar 

  132. A.N. Parshin, I.R. Shafarevich (eds.), Number Theory, vol. I, Introduction to Number Theory. Encyclopedia of Mathematical Sciences, vol. 49 (Springer, Berlin, 1995). Written by Yu. I. Manin and A. A. Panchishkin.

    Google Scholar 

  133. S.J. Patterson, An Introduction to the Theory of the Riemann Zeta-Function (Cambridge University Press, Cambridge, 1988)

    Book  MATH  Google Scholar 

  134. E.P.J. Pearse, Canonical self-affine tilings by iterated function systems. Indiana Univ. Math. J. 56(6), 3151–3169 (2007). Also: e-print, arXiv:math.MG/0606111, 2006

    Google Scholar 

  135. E.P.J. Pearse, S. Winter, Geometry of canonical self-similar tilings. Rocky Mountain J. Math. 42, 1327–1357 (2012). Also: e-print, arXiv:0811.2187, 2009

    Google Scholar 

  136. Ch. Pommerenke, Boundary Behavior of Conformal Maps (Springer, New York, 1992)

    Book  Google Scholar 

  137. A.G. Postnikov, Tauberian Theory and its Applications. Proceedings of the Steklov Institute of Mathematics (English transl., issue 2, 1980), vol. 144, 1979 (American Mathematical Society, Providence, RI, 1980)

    Google Scholar 

  138. C.R. Putnam, On the non-periodicity of the zeros of the Riemann zeta-function. Am. J. Math. 76, 97–99 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  139. C.R. Putnam, Remarks on periodic sequences and the Riemann zeta-function. Am. J. Math. 76, 828–830 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  140. G. Radunović, Fractal analysis of unbounded sets in Euclidean spaces and Lapidus zeta functions. Ph.D. thesis, University of Zagreb, Zagreb, Croatia, 2015

    Google Scholar 

  141. R. Rammal, Spectrum of harmonic excitations on fractals. J. Phys. 45, 191–206 (1984)

    Article  MathSciNet  Google Scholar 

  142. R. Rammal, G. Toulouse, Random walks on fractal structures and percolation cluster. J. Phys. Lett. 44, L13–L22 (1983)

    Article  Google Scholar 

  143. J. Rataj, S. Winter, On volume and surface area of parallel sets. Indiana Univ. Math. J. 59, 1661–1685 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  144. J. Rataj, S. Winter, Characterization of Minkowski measurability in terms of surface area. J. Math. Anal. Appl. 400, 120–132 (2013). Also: e-print, arXiv:1111.1825v2 [math.CA], 2012

    Google Scholar 

  145. M. Reed, B. Simon, Methods of Modern Mathematical Physics. Fourier Analysis, Self-Adjointness, vol. II (Academic, New York, 1975)

    Google Scholar 

  146. M. Reed, B. Simon, Methods of Modern Mathematical Physics. Analysis of Operators, vol. IV (Academic, New York, 1979)

    Google Scholar 

  147. M. Reed, B. Simon, Methods of Modern Mathematical Physics. Functional Analysis, vol. I, revised and enlarged edition (of the 1975 edn.) (Academic, New York, 1980)

    Google Scholar 

  148. A. Reich, Universelle Wertevereteilung von Eulerprodukten. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II(1), 1–17 (1977)

    Google Scholar 

  149. A. Reich, Wertverteilung von Zetafunktionen. Arch. Math. 34, 440–451 (1980)

    Article  MATH  Google Scholar 

  150. B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse. Monatsb. der Berliner Akad. pp. 671–680, 1858/1860. English transl. in [29], Appendix, pp. 229–305

    Google Scholar 

  151. W. Rudin, Real and Complex Analysis, 3rd edn. (McGraw-Hill, New York, 1987)

    MATH  Google Scholar 

  152. W. Rudin, Functional Analysis, 2nd edn. (of the 1973 edn.) (McGraw-Hill, New York, 1991)

    Google Scholar 

  153. C. Sabot, Integrated density of states of self-similar Sturm-Liouville operators and holomorphic dynamics in higher dimension. Ann. Inst. Henri Poincaré Probab. Stat. 37, 275–311 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  154. C. Sabot, Spectral properties of self-similar lattices and iteration of rational maps Mém. Soc. Math. Fr. (New Series) 92, 1–104 (2003)

    Google Scholar 

  155. C. Sabot, Spectral analysis of a self-similar Sturm-Liouville operator. Indiana Univ. Math. J. 54, 645–668 (2005)

    Article  MathSciNet  Google Scholar 

  156. B. Sapoval, Th. Gobron, A. Margolina, Vibrations of fractal drums. Phys. Rev. Lett. 67, 2974–2977 (1991)

    Article  Google Scholar 

  157. P. Sarnak, L-functions, in Proceedings of International Congress of Mathematicians, Berlin, 1998, ed. by G. Fischer, U. Rehmann, vol. I, pp. 453–465 (1998). Documenta Mathematica Journal DMV (Extra Volume ICM 98)

    Google Scholar 

  158. M. Schechter, Operator Methods in Quantum Mechanics (Dover Publications, Mineola, 2003)

    Google Scholar 

  159. M.R. Schroeder, Fractal, Chaos, Power Laws: Minutes From an Infinite Paradise (W. H. Freeman, New York, 1991)

    Google Scholar 

  160. L. Schwartz, Méthodes Mathématiques pour les Sciences Physiques (Hermann, Paris, 1961)

    MATH  Google Scholar 

  161. L. Schwartz, Théorie des Distributions, revised and enlarged edition (of the 1951 edn.) (Hermann, Paris, 1996)

    Google Scholar 

  162. R.T. Seeley, Complex Powers of Elliptic Operators. Proceedings of Symposia in Pure Mathematics, vol. 10 (American Mathematical Society, Providence, RI, 1967), pp. 288–307

    Google Scholar 

  163. R.T. Seeley, A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of \(\mathbb{R}^{3}\). Adv. Math. 29, 244–269 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  164. R.T. Seeley, An estimate near the boundary for the spectral counting function of the Laplace operator. Amer. J. Math. 102, 869–902 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  165. J.-P. Serre, A Course in Arithmetic, English transl. (Springer, Berlin, 1973)

    Book  Google Scholar 

  166. T. Shima, On eigenvalue problems for Laplacians on p.c.f. self-similar sets. Jpn. J. Ind. Appl. Math. 13, 1–23 (1996)

    Google Scholar 

  167. M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Ann. Math. 147, 225–267 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  168. B. Simon, Functional Integration and Quantum Physics (Academic, New York, 1979)

    MATH  Google Scholar 

  169. L.L. Stachó, On the volume function of parallel sets. Acta Sci. Math. 38, 365–374 (1976)

    MATH  Google Scholar 

  170. J. Steuding, Universality in the Selberg class, Special Activity in Analytic Number Theory and Diophantine Equations, in Proceedings of a Workshop Held at the Max Planck-Institut in Bonn, ed. by R.B. Heath-Brown, B. Moroz (2002). Bonner Math. Schriften 360, 2003

    Google Scholar 

  171. J. Steuding, Value-Distribution and L-Functions. Lecture Notes in Mathematics, vol. 1877 (Springer, Berlin, 2007)

    Google Scholar 

  172. A. Teplyaev, Spectral zeta functions of symmetric fractals, in Progress in Probability, vol. 57 (Birkhäuser-Verlag, Basel, 2004), pp. 245–262

    Google Scholar 

  173. A. Teplyaev, Spectral zeta functions of fractals and the complex dynamics of polynomials. Trans. Amer. Math. Soc. 359, 4339–4358 (2007). Also: e-print, arXiv:math.SP/ 0505546, 2005

    Google Scholar 

  174. E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd edn. (revised by D.R. Heath-Brown) (Oxford University Press, Oxford, 1986)

    Google Scholar 

  175. C. Tricot, Curves and Fractal Dimension (Springer, Berlin, 1995)

    Book  MATH  Google Scholar 

  176. M. van den Berg, P.B. Gilkey, A comparison estimate for the heat equation with an application to the heat content of the s-adic von Koch snowflake. Bull. Lond. Math. Soc. 30(4), 404–412 (1998)

    Article  MATH  Google Scholar 

  177. S.M. Voronin, The distribution of the non-zero values of the Riemann zeta function. Izv. Akad. Nauk. Inst. Steklov 128, 131–150 (1972) (Russian)

    MathSciNet  Google Scholar 

  178. S.M. Voronin, Theorem on the ‘universality’ of the Riemann zeta-function. Izv. Akad. Nauk. SSSR Ser. Matem. 39, 475–486 (1975) (Russian). Math. USSR Izv. 9 (1975), 443–445

    Google Scholar 

  179. H. Weyl, Hermann Weyl: Gesammelte Abhandlungen (Collected Works) (Springer, Berlin/New York, 1968)

    Google Scholar 

  180. H. Weyl, Über die Abhängigkeit der Eigenschwingungen einer Membran von deren Begrenzung. J. Reine Angew. Math. 141, 1–11 (1912). Reprinted in [179, vol. I, pp. 431–441]

    Google Scholar 

  181. H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann. 71, 441–479 (1912). Reprinted in [179, vol. I, pp. 393–430]

    Google Scholar 

  182. K.G. Wilson, Renormalization group and critical phenomena, I & II. Phys. Rev. B4, 3174–3183 & 3184–3205 (1971)

    Google Scholar 

  183. D. Z̆ubrinić, Minkowski content and singular integrals. Chaos Solitons Fractals 17(1), 169–177 (2003)

    Google Scholar 

  184. D. Z̆ubrinić, Analysis of Minkowski contents of fractal sets and applications. Real Anal. Exch. 31(2), 315–354 (2005/2006)

    Google Scholar 

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Acknowledgements

This research was partially supported by the US National Science Foundation (NSF) under the grants DMS-0707524 and DMS-1107750 (as well as by many earlier NSF grants since the mid-1980s). Part of this work was completed during several stays of the author as a visiting professor at the Institut des Hautes Etudes Scientifiques (IHES) in Paris/Bures-sur-Yvette, France.

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    Michel L. Lapidus

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    Carl Pomerance

  2. Department of Mathematics, ETH-Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Lapidus, M.L. (2015). The Sound of Fractal Strings and the Riemann Hypothesis. In: Pomerance, C., Rassias, M. (eds) Analytic Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-22240-0_14

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