Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (Springer Monographs in Mathematics)

Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. Key Features of this Second Edition: The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings Complex dimensions of a fractal string, defined as the poles of an associated zeta function, are studied in detail, then used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula The method of Diophantine approximation is used to study self-similar strings and flows Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions Throughout, new results are examined and a new definition of fractality as the presence of nonreal complex dimensions with positive real parts is presented. The new final chapter discusses several new topics and results obtained since the publication of the first edition. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. __Fractal Geometry, Complex Dimensions and Zeta Functions,__ Second Edition will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics. Lapidus......Page 1 Lapidus0......Page 2 Fractal Geometry, Complex Dimensions and Zeta Functions......Page 4 Overview......Page 8 Preface......Page 12 Contents......Page 16 List of Figures......Page 22 List of Tables......Page 26 Introduction......Page 28 1.1 The Geometry of a Fractal String......Page 36 1.1.1 The Multiplicity of the Lengths......Page 39 1.1.2 Example: The Cantor String......Page 40 1.2 The Geometric Zeta Function of a Fractal String......Page 43 1.2.1 The Screen and the Window......Page 46 1.2.2 The Cantor String (continued)......Page 49 1.3 The Frequencies of a Fractal String and the Spectral Zeta Function......Page 51 1.4 Higher-Dimensional Analogue: Fractal Sprays......Page 54 1.5 Notes......Page 57 2.1 Construction of a Self-Similar Fractal String......Page 60 2.1.1 Relation with Self-Similar Sets......Page 62 2.2 The Geometric Zeta Function of a Self-Similar String......Page 65 2.2.1 Self-Similar Strings with a Single Gap......Page 67 2.3.1 The Cantor String......Page 68 2.3.2 The Fibonacci String......Page 70 2.3.3 The Modified Cantor and Fibonacci Strings......Page 73 2.3.4 A String with Multiple Poles......Page 74 The Golden String......Page 76 2.4 The Lattice and Nonlattice Case......Page 79 2.5 The Structure of the Complex Dimensions......Page 81 2.6 The Asymptotic Density of the Poles in the Nonlattice Case......Page 88 2.7 Notes......Page 89 3 Complex Dimensions of Nonlattice Self-Similar Strings: Quasiperiodic Patterns and Diophantine Approximation......Page 91 3.1 Dirichlet Polynomial Equations......Page 92 3.1.1 The Generic Nonlattice Case......Page 93 3.2.1 Generic and Nongeneric Nonlattice Equations......Page 94 3.2.2 The Complex Roots of the Golden Plus Equation......Page 99 3.3 The Structure of the Complex Roots......Page 101 3.4 Approximating a Nonlattice Equation by Lattice Equations......Page 106 3.4.1 Diophantine Approximation......Page 109 3.4.2 The Quasiperiodic Pattern of the Complex Dimensions......Page 112 3.4.3 Application to Nonlattice Strings......Page 115 3.5 Complex Roots of a Nonlattice Dirichlet Polynomial......Page 118 3.5.1 Continued Fractions......Page 119 3.5.2 Two Generators......Page 120 3.5.3 More than Two Generators......Page 126 3.6 Dimension-Free Regions......Page 129 3.7 The Dimensions of Fractality of a Nonlattice String......Page 136 3.7.1 The Density of the Real Parts......Page 137 3.8 A Note on the Computations......Page 141 3.9 Notes......Page 143 4 Generalized Fractal Strings Viewed as Measures......Page 144 4.1 Generalized Fractal Strings......Page 145 4.1.1 Examples of Generalized Fractal Strings......Page 147 4.2 The Frequencies of a Generalized Fractal String......Page 149 4.2.1 Completion of the Harmonic String: Euler Product......Page 152 4.3 Generalized Fractal Sprays......Page 154 4.4 The Measure of a Self-Similar String......Page 155 4.4.1 Measures with a Self-Similarity Property......Page 157 4.5 Notes......Page 160 5.1 Introduction......Page 161 5.1.1 Outline of the Proof......Page 163 5.1.2 Examples......Page 164 5.2 Preliminaries: The Heaviside Function......Page 166 5.3 Pointwise Explicit Formulas......Page 170 5.3.1 The Order of Growth of the Sum over the Complex Dimensions......Page 181 5.4 Distributional Explicit Formulas......Page 182 5.4.1 Extension to More General Test Functions......Page 187 5.4.2 The Order of the Distributional Error Term......Page 192 5.5 Example: The Prime Number Theorem......Page 198 5.5.1 The Riemann–von Mangoldt Formula......Page 200 5.6 Notes......Page 201 6 The Geometry and the Spectrum of Fractal Strings......Page 203 6.1.1 The Geometric Local Terms......Page 204 6.1.2 The Spectral Local Terms......Page 206 6.1.4 The Distribution xω log mx......Page 207 6.2.1 The Geometric Counting Function of a Fractal String......Page 208 6.2.2 The Spectral Counting Function of a Fractal String......Page 209 6.2.3 The Geometric and Spectral Partition Functions......Page 211 6.3.1 The Density of Geometric and Spectral States......Page 213 6.3.2 The Spectral Operator and its Euler Product......Page 214 6.4 Self-Similar Strings......Page 218 6.4.1 Lattice Strings......Page 219 6.4.2 Nonlattice Strings......Page 222 Lattice Case......Page 224 Nonlattice Case......Page 226 6.5.1 The a-String......Page 227 6.5.2 The Spectrum of the Harmonic String......Page 230 6.6 Fractal Sprays......Page 231 6.6.1 The Sierpinski Drum......Page 232 6.6.2 The Spectrum of a Self-Similar Spray......Page 236 7 Periodic Orbits of Self-Similar Flows......Page 237 7.1 Suspended Flows......Page 238 7.1.1 The Zeta Function of a Dynamical System......Page 239 7.2 Periodic Orbits, Euler Product......Page 241 7.3 Self-Similar Flows......Page 243 7.3.1 Examples of Self-Similar Flows......Page 247 7.3.2 The Lattice and Nonlattice Case......Page 248 7.4 The Prime Orbit Theorem for Suspended Flows......Page 250 7.4.2 Lattice Flows......Page 252 7.4.3 Nonlattice Flows......Page 253 7.5.1 Two Generators......Page 254 7.5.2 More Than Two Generators......Page 256 7.6 Notes......Page 259 8 Fractal Tube Formulas......Page 260 8.1 Explicit Formulas for the Volume of Tubular Neighborhoods......Page 261 8.1.1 The Pointwise Tube Formula......Page 266 8.1.2 Example: The a-String......Page 269 8.2 Analogy with Riemannian Geometry......Page 270 8.3 Minkowski Measurability and Complex Dimensions......Page 271 8.4.1 Generalized Cantor Strings......Page 276 8.4.2 Lattice Self-Similar Strings......Page 279 8.4.3 The Average Minkowski Content......Page 284 8.4.4 Nonlattice Self-Similar Strings......Page 287 8.5 Notes......Page 291 9 Riemann Hypothesis and Inverse Spectral Problems......Page 294 9.1 The Inverse Spectral Problem......Page 295 9.2 Complex Dimensions and the Riemann Hypothesis......Page 298 9.3 Fractal Sprays and the Generalized Riemann Hypothesis......Page 301 9.4 Notes......Page 303 10.1 The Geometry of a Generalized Cantor String......Page 305 10.2 The Spectrum of a Generalized Cantor String......Page 308 10.2.1 Integral Cantor Strings: a-adic Analysis of the Geometric and Spectral Oscillations......Page 310 10.3 The Truncated Cantor String......Page 313 10.3.1 The Spectrum of the Truncated Cantor String......Page 316 10.4 Notes......Page 317 11 Critical Zeros of Zeta Functions......Page 318 11.1 The Riemann Zeta Function: No Critical Zeros in Arithmetic Progression......Page 319 11.1.1 Finite Arithmetic Progressions of Zeros......Page 322 11.2 Extension to Other Zeta Functions......Page 328 11.3 Density of Nonzeros on Vertical Lines......Page 330 11.3.1 Almost Arithmetic Progressions of Zeros......Page 331 11.4 Extension to L-Series......Page 332 11.4.1 Finite Arithmetic Progressions of Zeros of L-Series......Page 333 The Direct Computation......Page 336 The Explicit Formula......Page 337 Zeros in Arithmetic Progression......Page 339 11.5 Conjectures about Zeros of Dirichlet Series......Page 341 11.6 Zeta Functions of Curves Over Finite Fields......Page 345 12 Fractality and Complex Dimensions......Page 354 12.1 A New Definition of Fractality......Page 355 12.1.1 Fractal Geometers’ Intuition of Fractality......Page 356 12.1.2 Our Definition of Fractality......Page 358 12.1.3 Possible Connections with the Notion of Lacunarity......Page 363 12.2 Fractality and Self-Similarity......Page 365 12.2.1 Complex Dimensions and Tube Formula for the von Koch Snowflake Curve......Page 367 12.3 Complex Dimensions and Shifts......Page 374 12.4 Complex Dimensions as Geometric Invariants......Page 375 12.4.1 Connection with Varieties over Finite Fields......Page 377 Fractal Geometries and Finite Geometries......Page 378 12.5.1 The Weyl–Berry Conjecture......Page 381 12.5.2 The Spectrum of a Self-Similar Drum......Page 382 12.5.3 Spectrum and Periodic Orbits......Page 386 12.6 Notes......Page 391 13 New Results and Perspectives......Page 393 13.1 A Higher-Dimensional Theory of Complex Dimensions......Page 394 13.1.1 The Distributional Tube Formula for Fractal Sprays with Monophase Generators......Page 397 13.1.2 The Canonical Self-Similar Tiling......Page 403 13.1.3 Convex Geometry and the Curvature Measures......Page 409 13.1.4 Examples of Tube Formulas for Fractal Sprays and Self-Similar Tilings......Page 413 13.1.5 Pointwise Tube Formula for Fractal Sprays......Page 418 13.2 p-Adic Fractal Strings......Page 430 13.2.1 p-Adic Numbers......Page 431 13.2.2 p-Adic Fractal Strings......Page 433 13.2.3 Inner Tube and its Volume......Page 437 13.2.4 Tube Formulas for p-Adic Fractal Strings......Page 442 13.2.5 Self-Similar p-Adic Fractal Strings......Page 446 13.2.6 The Geometric Zeta Function......Page 450 13.2.7 Periodicity of the Poles and the Zeros......Page 451 13.2.8 Exact Tube Formula for p-Adic Self-Similar Strings......Page 454 13.2.9 Tube Formula for the 2-Adic Fibonacci String......Page 457 13.2.10 The 3-Adic Cantor Set......Page 459 13.2.11 The Minkowski Dimension......Page 461 13.2.12 The Average Minkowski Content......Page 465 13.2.13 Concluding Comments......Page 466 13.3 Multifractal Analysis via Zeta Functions......Page 467 13.3.1 Regularity......Page 469 13.3.2 Multifractal Zeta Functions......Page 470 13.3.3 Results for Fractal Strings......Page 472 13.3.4 Variants of the Cantor String......Page 475 13.3.5 A Full Collection of Multifractal Zeta Functions......Page 478 13.3.6 A Classic Multifractal and Other Zeta Functions......Page 482 13.4 Random Fractal Strings and their Zeta Functions......Page 487 13.4.1 Random Self-Similar Strings......Page 490 13.4.2 Stable Random Strings......Page 491 13.4.3 A Refinement of the Notion of Fractality......Page 493 13.5 Fractal Membranes: Quantized Fractal Strings......Page 494 13.5.1 Noncommutative, Infinite Dimensional Tori......Page 495 Fractal Membranes and their Zeta Functions......Page 497 13.5.2 Construction of Fractal Membranes......Page 499 13.5.3 Flows on the Moduli Space of Fractal Membranes......Page 501 13.6 Notes......Page 503 A.1 The Dedekind Zeta Function......Page 504 A.2 Characters and Hecke L-series......Page 505 A.3 Completion of L-Series, Functional Equation......Page 506 A.4 Epstein Zeta Functions......Page 507 A.5 Two-Variable Zeta Functions......Page 508 A.5.1 The Zeta Function of Pellikaan......Page 509 A.5.2 The Zeta Function of Schoof and van der Geer......Page 511 A.6 Other Zeta Functions in Number Theory......Page 514 B.1 Weyl’s Asymptotic Formula......Page 515 B.2 Heat Asymptotic Expansion......Page 517 B.3 The Spectral Zeta Function and its Poles......Page 518 B.4 Extensions......Page 520 B.4.1 Monotonic Second Term......Page 521 B.5 Notes......Page 522 Appendix C An Application of Nevanlinna Theory......Page 523 C.1 The Nevanlinna Height......Page 524 C.2 Complex Zeros of Dirichlet Polynomials......Page 525 Acknowledgements......Page 529 Bibliography......Page 533 Author Index......Page 566 Subject Index......Page 569 Index of Symbols......Page 579 Conventions......Page 583 Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings; that is, one-dimensional drums with fractal boundary. This second edition of Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, complex analysis, distribution theory, and mathematical physics. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Key Features include: · The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings · Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition: " The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actually has many applications." —Nicolae-Adrian Secelean, Zentralblatt Key Features include: · The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings · Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition: " The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actually has many applications." —Nicolae-Adrian Secelean, Zentralblatt · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition: " The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actually has many applications." —Nicolae-Adrian Secelean, Zentralblatt Key Features include: · The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings · Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition: " The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actually has many applications." —Nicolae-Adrian Secelean, Zentralblatt · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition: " The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actually has many applications." —Nicolae-Adrian Secelean, Zentralblatt · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition: " The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actually has many applications." —Nicolae-Adrian Secelean, Zentralblatt Key Features include: · The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings · Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings; that is, one-dimensional drums with fractal boundary. This second edition of Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, complex analysis, distribution theory, and mathematical physics. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Key Features include:· The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings· Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra· Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal· Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula· The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functionsThe unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition:" The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ... The book is very well written and organized and the subject is very interesting and actually has many applications."--Nicolae-Adrian Secelean, Zentralblatt Key Features include:· The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings· Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra· Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal· Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula· The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functionsThe unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition:" The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ... The book is very well written and organized and the subject is very interesting and actually has many applications."--Nicolae-Adrian Secelean, Zentralblatt · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal· Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula· The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functionsThe unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition:" The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ... The book is very well written and organized and the subject is very interesting and actually has many applications."--Nicolae-Adrian Secelean, Zentralblatt Key Features include:· The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings· Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra· Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal· Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula· The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functionsThe unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition:" The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ... The book is very well written and organized and the subject is very interesting and actually has many applications." -- Nicolae-Adrian Secelean, Zentralblatt · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal· Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula· The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functionsThe unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition:" The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ... The book is very well written and organized and the subject is very interesting and actually has many applications." -- Nicolae-Adrian Secelean, Zentralblatt · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal· Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula· The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functionsThe unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition:" The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ... The book is very well written and organized and the subject is very interesting and actually has many applications." -- Nicolae-Adrian Secelean, Zentralblatt Key Features include:· The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings· Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra· Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal· Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula· The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-the Preface -- Overview -- Introduction -- 1. Complex Dimensions Of Ordinary Fractal Strings -- 2. Complex Dimensions Of Self-similar Fractal Strings -- 3. Complex Dimensions Of Nonlattice Self-similar Strings -- 4. Generalized Fractal Strings Viewed As Measures -- 5. Explicit Formulas For Generalized Fractal Strings -- 6. The Geometry And The Spectrum Of Fractal Strings -- 7. Periodic Orbits Of Self-similar Flows -- 8. Fractal Tube Formulas -- 9. Riemann Hypothesis And Inverse Spectral Problems -- 10. Generalized Cantor Strings And Their Oscillations -- 11. Critical Zero Of Zeta Functions -- 12 Fractality And Complex Dimensions -- 13. Recent Results And Perspectives -- Appendix A. Zeta Functions In Number Theory -- Appendix B. Zeta Functions Of Laplacians And Spectral Asymptotics -- Appendix C. An Application Of Nevanlinna Theory -- Bibliography -- Author Index -- Subject Index -- Index Of Symbols -- Conventions -- Acknowledgements. Michel L. Lapidus, Machiel Van Frankenhuijsen. Includes Bibliographical References (p. 515-547) And Index. Complex Dimensions of Ordinary Fractal Strings -- Complex Dimensions of Self-Similar Fractal Strings -- Complex Dimensions of Nonlattice Self-Similar Strings: Quasiperiodic Patterns and Diophantine Approximation -- Generalized Fractal Strings Viewed as Measures -- Explicit Formulas for Generalized Fractal Strings -- The Geometry and the Spectrum of Fractal Strings -- Periodic Orbits of Self-Similar Flows -- Fractal Tube Formulas -- Riemann Hypothesis and Inverse Spectral Problems -- Generalized Cantor Strings and their Oscillations -- Critical Zeros of Zeta Functions -- Fractality and Complex Dimensions -- New Results and Perspectives