Spectral Theory Of The Riemann Zeta Function

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Spectral Theory Of The Riemann Zeta Function

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Author : Yoichi Motohashi
language : en
Publisher: Cambridge University Press
Release Date : 1997-09-11

Spectral Theory Of The Riemann Zeta Function written by Yoichi Motohashi and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-09-11 with Mathematics categories.

The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. In this book, based on his own research, Professor Motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilized to unveil an image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the development of number theory, namely the fusion of two main fields in mathematics that were previously studied separately.

Spectral Theory Of The Riemann Zeta Function

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Author : Yoichi Motohashi
language : en
Publisher: Cambridge University Press
Release Date : 1997-09-11

Spectral Theory Of The Riemann Zeta Function written by Yoichi Motohashi and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-09-11 with Mathematics categories.

The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. In this book, based on his own research, Professor Motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilized to unveil an image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the development of number theory, namely the fusion of two main fields in mathematics that were previously studied separately.

Spectral Theory Of Automorphic Functions

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Author : A. B. Venkov
language : en
Publisher: American Mathematical Soc.
Release Date : 1983

Spectral Theory Of Automorphic Functions written by A. B. Venkov and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1983 with Mathematics categories.

Riemann Hypothesis And Spectral Theory

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Author : Jason Cole
language : en
Publisher:
Release Date : 2020-01-10

Riemann Hypothesis And Spectral Theory written by Jason Cole and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-01-10 with categories.

This book provides a brief overview of the Riemann Zeta function, Riemann Hypothesis and the Hilbert-Polya spectral operator approach to proving RH. Also included in this book is a new discovery that describes a correlation between the Riemann Xi function and gravity rotational curves. Surprisingly their is a mathematical correlation between the complex system of the Riemann Xi function and the large scale distribution of galaxies and rotational curves. Also included in this book are new discoveries on the Prime Number theorem, Riemann Zeta function and other new science and math discoveries.

An Approach To The Selberg Trace Formula Via The Selberg Zeta Function

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Author : Jürgen Fischer
language : en
Publisher: Springer
Release Date : 2006-11-15

An Approach To The Selberg Trace Formula Via The Selberg Zeta Function written by Jürgen Fischer and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-15 with Mathematics categories.

The Notes give a direct approach to the Selberg zeta-function for cofinite discrete subgroups of SL (2,#3) acting on the upper half-plane. The basic idea is to compute the trace of the iterated resolvent kernel of the hyperbolic Laplacian in order to arrive at the logarithmic derivative of the Selberg zeta-function. Previous knowledge of the Selberg trace formula is not assumed. The theory is developed for arbitrary real weights and for arbitrary multiplier systems permitting an approach to known results on classical automorphic forms without the Riemann-Roch theorem. The author's discussion of the Selberg trace formula stresses the analogy with the Riemann zeta-function. For example, the canonical factorization theorem involves an analogue of the Euler constant. Finally the general Selberg trace formula is deduced easily from the properties of the Selberg zeta-function: this is similar to the procedure in analytic number theory where the explicit formulae are deduced from the properties of the Riemann zeta-function. Apart from the basic spectral theory of the Laplacian for cofinite groups the book is self-contained and will be useful as a quick approach to the Selberg zeta-function and the Selberg trace formula.

Dynamical Spectral And Arithmetic Zeta Functions

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Author : Michel Laurent Lapidus
language : en
Publisher: American Mathematical Soc.
Release Date : 2001

Dynamical Spectral And Arithmetic Zeta Functions written by Michel Laurent Lapidus and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.

The original zeta function was studied by Riemann as part of his investigation of the distribution of prime numbers. Other sorts of zeta functions were defined for number-theoretic purposes, such as the study of primes in arithmetic progressions. This led to the development of $L$-functions, which now have several guises. It eventually became clear that the basic construction used for number-theoretic zeta functions can also be used in other settings, such as dynamics, geometry, and spectral theory, with remarkable results. This volume grew out of the special session on dynamical, spectral, and arithmetic zeta functions held at the annual meeting of the American Mathematical Society in San Antonio, but also includes four articles that were invited to be part of the collection. The purpose of the meeting was to bring together leading researchers, to find links and analogies between their fields, and to explore new methods. The papers discuss dynamical systems, spectral geometry on hyperbolic manifolds, trace formulas in geometry and in arithmetic, as well as computational work on the Riemann zeta function. Each article employs techniques of zeta functions. The book unifies the application of these techniques in spectral geometry, fractal geometry, and number theory. It is a comprehensive volume, offering up-to-date research. It should be useful to both graduate students and confirmed researchers.

Spectral Theory Of Infinite Area Hyperbolic Surfaces

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Author : David Borthwick
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-09-13

Spectral Theory Of Infinite Area Hyperbolic Surfaces written by David Borthwick and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-09-13 with Mathematics categories.

This book is a self-contained monograph on spectral theory for non-compact Riemann surfaces, focused on the infinite-volume case. By focusing on the scattering theory of hyperbolic surfaces, this work provides a compelling introductory example which will be accessible to a broad audience. The book opens with an introduction to the geometry of hyperbolic surfaces. Then a thorough development of the spectral theory of a geometrically finite hyperbolic surface of infinite volume is given. The final sections include recent developments for which no thorough account exists.

Fractal Geometry Complex Dimensions And Zeta Functions

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Author : Michel Lapidus
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-09-20

Fractal Geometry Complex Dimensions And Zeta Functions written by Michel Lapidus and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-09-20 with Mathematics categories.

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. Throughout Geometry, Complex Dimensions and Zeta Functions, Second Edition, 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.

Zeta Functions Of Graphs

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Author : Audrey Terras
language : en
Publisher: Cambridge University Press
Release Date : 2010-11-18

Zeta Functions Of Graphs written by Audrey Terras and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-11-18 with Mathematics categories.

Graph theory meets number theory in this stimulating book. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann/Dedekind zeta functions. For example, there is a Riemann hypothesis (which may be false) and prime number theorem for graphs. Explicit constructions of graph coverings use Galois theory to generalize Cayley and Schreier graphs. Then non-isomorphic simple graphs with the same zeta are produced, showing you cannot hear the shape of a graph. The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book, which makes connections with quantum chaos and random matrix theory, plus expander/Ramanujan graphs of interest in computer science. Created for beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and exercises, both theoretical and computer-based, are included throughout.

An Introduction To The Theory Of The Riemann Zeta Function

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Author : S. J. Patterson
language : en
Publisher: Cambridge University Press
Release Date : 1995-02-02

An Introduction To The Theory Of The Riemann Zeta Function written by S. J. Patterson and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995-02-02 with Mathematics categories.

An introduction to the analytic techniques used in the investigation of zeta functions through the example of the Riemann zeta function. It emphasizes central ideas of broad application, avoiding technical results and the customary function-theoretic appro