Fractal Geometry Complex Dimensions and Zeta Functions : Geometry and Spectra of Fractal Strings 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. 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.