Download or read book Riemann Hypothesis and Spectral Theory written by Jason Cole and published by . This book was released on 2021-09-19 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Spectral Theory of the Riemann Zeta Function written by Yoichi Motohashi and published by Cambridge University Press. This book was released on 1997-09-11 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Quantized Number Theory Fractal Strings And The Riemann Hypothesis From Spectral Operators To Phase Transitions And Universality written by Hafedh Herichi and published by World Scientific. This book was released on 2021-07-27 with total page 494 pages. Available in PDF, EPUB and Kindle. Book excerpt: Studying the relationship between the geometry, arithmetic and spectra of fractals has been a subject of significant interest in contemporary mathematics. This book contributes to the literature on the subject in several different and new ways. In particular, the authors provide a rigorous and detailed study of the spectral operator, a map that sends the geometry of fractal strings onto their spectrum. To that effect, they use and develop methods from fractal geometry, functional analysis, complex analysis, operator theory, partial differential equations, analytic number theory and mathematical physics.Originally, M L Lapidus and M van Frankenhuijsen 'heuristically' introduced the spectral operator in their development of the theory of fractal strings and their complex dimensions, specifically in their reinterpretation of the earlier work of M L Lapidus and H Maier on inverse spectral problems for fractal strings and the Riemann hypothesis.One of the main themes of the book is to provide a rigorous framework within which the corresponding question 'Can one hear the shape of a fractal string?' or, equivalently, 'Can one obtain information about the geometry of a fractal string, given its spectrum?' can be further reformulated in terms of the invertibility or the quasi-invertibility of the spectral operator.The infinitesimal shift of the real line is first precisely defined as a differentiation operator on a family of suitably weighted Hilbert spaces of functions on the real line and indexed by a dimensional parameter c. Then, the spectral operator is defined via the functional calculus as a function of the infinitesimal shift. In this manner, it is viewed as a natural 'quantum' analog of the Riemann zeta function. More precisely, within this framework, the spectral operator is defined as the composite map of the Riemann zeta function with the infinitesimal shift, viewed as an unbounded normal operator acting on the above Hilbert space.It is shown that the quasi-invertibility of the spectral operator is intimately connected to the existence of critical zeros of the Riemann zeta function, leading to a new spectral and operator-theoretic reformulation of the Riemann hypothesis. Accordingly, the spectral operator is quasi-invertible for all values of the dimensional parameter c in the critical interval (0,1) (other than in the midfractal case when c =1/2) if and only if the Riemann hypothesis (RH) is true. A related, but seemingly quite different, reformulation of RH, due to the second author and referred to as an 'asymmetric criterion for RH', is also discussed in some detail: namely, the spectral operator is invertible for all values of c in the left-critical interval (0,1/2) if and only if RH is true.These spectral reformulations of RH also led to the discovery of several 'mathematical phase transitions' in this context, for the shape of the spectrum, the invertibility, the boundedness or the unboundedness of the spectral operator, and occurring either in the midfractal case or in the most fractal case when the underlying fractal dimension is equal to ½ or 1, respectively. In particular, the midfractal dimension c=1/2 is playing the role of a critical parameter in quantum statistical physics and the theory of phase transitions and critical phenomena.Furthermore, the authors provide a 'quantum analog' of Voronin's classical theorem about the universality of the Riemann zeta function. Moreover, they obtain and study quantized counterparts of the Dirichlet series and of the Euler product for the Riemann zeta function, which are shown to converge (in a suitable sense) even inside the critical strip.For pedagogical reasons, most of the book is devoted to the study of the quantized Riemann zeta function. However, the results obtained in this monograph are expected to lead to a quantization of most classic arithmetic zeta functions, hence, further 'naturally quantizing' various aspects of analytic number theory and arithmetic geometry.The book should be accessible to experts and non-experts alike, including mathematics and physics graduate students and postdoctoral researchers, interested in fractal geometry, number theory, operator theory and functional analysis, differential equations, complex analysis, spectral theory, as well as mathematical and theoretical physics. Whenever necessary, suitable background about the different subjects involved is provided and the new work is placed in its proper historical context. Several appendices supplementing the main text are also included.

Download or read book Spectral Theory of the Riemann Zeta Function written by Yoichi Motohashi and published by Cambridge University Press. This book was released on 1997-09-11 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book The Riemann Hypothesis written by Peter B. Borwein and published by Springer Science & Business Media. This book was released on 2008 with total page 543 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Riemann Hypothesis has become the Holy Grail of mathematics in the century and a half since 1859 when Bernhard Riemann, one of the extraordinary mathematical talents of the 19th century, originally posed the problem. While the problem is notoriously difficult, and complicated even to state carefully, it can be loosely formulated as "the number of integers with an even number of prime factors is the same as the number of integers with an odd number of prime factors." The Hypothesis makes a very precise connection between two seemingly unrelated mathematical objects, namely prime numbers and the zeros of analytic functions. If solved, it would give us profound insight into number theory and, in particular, the nature of prime numbers. This book is an introduction to the theory surrounding the Riemann Hypothesis. Part I serves as a compendium of known results and as a primer for the material presented in the 20 original papers contained in Part II. The original papers place the material into historical context and illustrate the motivations for research on and around the Riemann Hypothesis. Several of these papers focus on computation of the zeta function, while others give proofs of the Prime Number Theorem, since the Prime Number Theorem is so closely connected to the Riemann Hypothesis. The text is suitable for a graduate course or seminar or simply as a reference for anyone interested in this extraordinary conjecture.

Download or read book An Introduction to the Theory of the Riemann Zeta Function written by S. J. Patterson and published by Cambridge University Press. This book was released on 1995-02-02 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Download or read book In Search of the Riemann Zeros written by Michel Laurent Lapidus and published by American Mathematical Soc.. This book was released on 2008 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: Formulated in 1859, the Riemann Hypothesis is the most celebrated and multifaceted open problem in mathematics. In essence, it states that the primes are distributed as harmoniously as possible--or, equivalently, that the Riemann zeros are located on a single vertical line, called the critical line.

Download or read book Zeta and L functions in Number Theory and Combinatorics written by Wen-Ching Winnie Li and published by American Mathematical Soc.. This book was released on 2019-03-01 with total page 95 pages. Available in PDF, EPUB and Kindle. Book excerpt: Zeta and L-functions play a central role in number theory. They provide important information of arithmetic nature. This book, which grew out of the author's teaching over several years, explores the interaction between number theory and combinatorics using zeta and L-functions as a central theme. It provides a systematic and comprehensive account of these functions in a combinatorial setting and establishes, among other things, the combinatorial counterparts of celebrated results in number theory, such as the prime number theorem and the Chebotarev density theorem. The spectral theory for finite graphs and higher dimensional complexes is studied. Of special interest in theory and applications are the spectrally extremal objects, called Ramanujan graphs and Ramanujan complexes, which can be characterized by their associated zeta functions satisfying the Riemann Hypothesis. Explicit constructions of these extremal combinatorial objects, using number-theoretic and combinatorial means, are presented. Research on zeta and L-functions for complexes other than graphs emerged only in recent years. This is the first book for graduate students and researchers offering deep insight into this fascinating and fast developing area.

Download or read book Spectral Theory of Automorphic Functions written by A.B. Venkov and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 189 pages. Available in PDF, EPUB and Kindle. Book excerpt: 'Et moi ..., si j'avait su comment en revcnrr, One service mathematics has rendered the je n'y serais point aile.' human race. It has put common sense back. Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Download or read book Geometry Spectral Theory Groups and Dynamics written by Robert Brooks and published by American Mathematical Soc.. This book was released on 2005 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains articles based on talks given at the Robert Brooks Memorial Conference on Geometry and Spectral Theory and the Workshop on Groups, Geometry and Dynamics held at Technion - the Israel Institute of Technology (Haifa). Robert Brooks' (1952 - 2002) broad range of mathematical interests is represented in the volume, which is devoted to various aspects of global analysis, spectral theory, the theory of Riemann surfaces, Riemannian and discrete geometry, and numbertheory. A survey of Brooks' work has been written by his close colleague, Peter Buser. Also included in the volume are articles on analytic topics, such as Szego's theorem, and on geometric topics, such as isoperimetric inequalities and symmetries of manifolds. The book is suitable for graduate studentsand researchers interested in various aspects of geometry and global analysis.

Download or read book Schr dinger Operators Spectral Analysis and Number Theory written by Sergio Albeverio and published by Springer Nature. This book was released on 2021-06-03 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives its readers a unique opportunity to get acquainted with new aspects of the fruitful interactions between Analysis, Geometry, Quantum Mechanics and Number Theory. The present book contains a number of contributions by specialists in these areas as an homage to the memory of the mathematician Erik Balslev and, at the same time, advancing a fascinating interdisciplinary area still full of potential. Erik Balslev has made original and important contributions to several areas of Mathematics and its applications. He belongs to the founders of complex scaling, one of the most important methods in the mathematical and physical study of eigenvalues and resonances of Schrödinger operators, which has been very essential in advancing the solution of fundamental problems in Quantum Mechanics and related areas. He was also a pioneer in making available and developing spectral methods in the study of important problems in Analytic Number Theory.

Download or read book Generalized Minkowski Content Spectrum of Fractal Drums Fractal Strings and the Riemann Zeta Functions written by Christina Q. He and published by American Mathematical Soc.. This book was released on 1997 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of "fractal drums" (and especially of "fractal strings"). In this work, the authors extend previous results in this area by using the notionof generalized Minkowski content which is defined through some suitable "gauge functions" other than power functions. (This content is used to measure the irregularity (or "fractality") of the boundary of an open set in R]n by evaluating the volume of its small tubular neighborhoods). In the situation when the power function is not the natural "gauge function", this enables the authors to obtain more precise estimates, with a broader potential range of applications than in previous papers of the second author and his collaborators. This text will also be of interest to those working in mathematical physics.

Download or read book Supersymmetry and Trace Formulae written by Igor V. Lerner and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: The motion of a particle in a random potential in two or more dimensions is chaotic, and the trajectories in deterministically chaotic systems are effectively random. It is therefore no surprise that there are links between the quantum properties of disordered systems and those of simple chaotic systems. The question is, how deep do the connec tions go? And to what extent do the mathematical techniques designed to understand one problem lead to new insights into the other? The canonical problem in the theory of disordered mesoscopic systems is that of a particle moving in a random array of scatterers. The aim is to calculate the statistical properties of, for example, the quantum energy levels, wavefunctions, and conductance fluctuations by averaging over different arrays; that is, by averaging over an ensemble of different realizations of the random potential. In some regimes, corresponding to energy scales that are large compared to the mean level spacing, this can be done using diagrammatic perturbation theory. In others, where the discreteness of the quantum spectrum becomes important, such an approach fails. A more powerful method, devel oped by Efetov, involves representing correlation functions in terms of a supersymmetric nonlinear sigma-model. This applies over a wider range of energy scales, covering both the perturbative and non-perturbative regimes. It was proved using this method that energy level correlations in disordered systems coincide with those of random matrix theory when the dimensionless conductance tends to infinity.

Download or read book Noncommutative Geometry Quantum Fields and Motives written by Alain Connes and published by American Mathematical Soc.. This book was released on 2019-03-13 with total page 785 pages. Available in PDF, EPUB and Kindle. Book excerpt: The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adèle class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.

Download or read book Fractal Geometry Complex Dimensions and Zeta Functions written by Michel Lapidus and published by Springer Science & Business Media. This book was released on 2012-09-20 with total page 583 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Functional Analysis Spectral Theory and Applications written by Manfred Einsiedler and published by Springer. This book was released on 2017-11-21 with total page 614 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook provides a careful treatment of functional analysis and some of its applications in analysis, number theory, and ergodic theory. In addition to discussing core material in functional analysis, the authors cover more recent and advanced topics, including Weyl’s law for eigenfunctions of the Laplace operator, amenability and property (T), the measurable functional calculus, spectral theory for unbounded operators, and an account of Tao’s approach to the prime number theorem using Banach algebras. The book further contains numerous examples and exercises, making it suitable for both lecture courses and self-study. Functional Analysis, Spectral Theory, and Applications is aimed at postgraduate and advanced undergraduate students with some background in analysis and algebra, but will also appeal to everyone with an interest in seeing how functional analysis can be applied to other parts of mathematics.

Download or read book Exploring the Riemann Zeta Function written by Hugh Montgomery and published by Springer. This book was released on 2017-09-11 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: Exploring the Riemann Zeta Function: 190 years from Riemann's Birth presents a collection of chapters contributed by eminent experts devoted to the Riemann Zeta Function, its generalizations, and their various applications to several scientific disciplines, including Analytic Number Theory, Harmonic Analysis, Complex Analysis, Probability Theory, and related subjects. The book focuses on both old and new results towards the solution of long-standing problems as well as it features some key historical remarks. The purpose of this volume is to present in a unified way broad and deep areas of research in a self-contained manner. It will be particularly useful for graduate courses and seminars as well as it will make an excellent reference tool for graduate students and researchers in Mathematics, Mathematical Physics, Engineering and Cryptography.