Zeta Functions Over Zeros Of Zeta Functions
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| Author | : Anatoly A. Karatsuba |
| Publisher | : Walter de Gruyter |
| Total Pages | : 409 |
| Release | : 2011-05-03 |
| Genre | : Mathematics |
| ISBN | : 3110886146 |
The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany
| Author | : Harold M. Edwards |
| Publisher | : Courier Corporation |
| Total Pages | : 338 |
| Release | : 2001-01-01 |
| Genre | : Mathematics |
| ISBN | : 9780486417400 |
Superb high-level study of one of the most influential classics in mathematics examines landmark 1859 publication entitled “On the Number of Primes Less Than a Given Magnitude,” and traces developments in theory inspired by it. Topics include Riemann's main formula, the prime number theorem, the Riemann-Siegel formula, large-scale computations, Fourier analysis, and other related topics. English translation of Riemann's original document appears in the Appendix.
| Author | : H. Iwaniec |
| Publisher | : American Mathematical Society |
| Total Pages | : 130 |
| Release | : 2014-10-07 |
| Genre | : Mathematics |
| ISBN | : 1470418517 |
The Riemann zeta function was introduced by L. Euler (1737) in connection with questions about the distribution of prime numbers. Later, B. Riemann (1859) derived deeper results about the prime numbers by considering the zeta function in the complex variable. The famous Riemann Hypothesis, asserting that all of the non-trivial zeros of zeta are on a critical line in the complex plane, is one of the most important unsolved problems in modern mathematics. The present book consists of two parts. The first part covers classical material about the zeros of the Riemann zeta function with applications to the distribution of prime numbers, including those made by Riemann himself, F. Carlson, and Hardy-Littlewood. The second part gives a complete presentation of Levinson's method for zeros on the critical line, which allows one to prove, in particular, that more than one-third of non-trivial zeros of zeta are on the critical line. This approach and some results concerning integrals of Dirichlet polynomials are new. There are also technical lemmas which can be useful in a broader context.
| Author | : Jörn Steuding |
| Publisher | : Springer |
| Total Pages | : 320 |
| Release | : 2007-05-26 |
| Genre | : Mathematics |
| ISBN | : 3540448225 |
These notes present recent results in the value-distribution theory of L-functions with emphasis on the phenomenon of universality. Universality has a strong impact on the zero-distribution: Riemann’s hypothesis is true only if the Riemann zeta-function can approximate itself uniformly. The text proves universality for polynomial Euler products. The authors’ approach follows mainly Bagchi's probabilistic method. Discussion touches on related topics: almost periodicity, density estimates, Nevanlinna theory, and functional independence.
| Author | : Aleksandar Ivic |
| Publisher | : Courier Corporation |
| Total Pages | : 548 |
| Release | : 2012-07-12 |
| Genre | : Mathematics |
| ISBN | : 0486140040 |
This text covers exponential integrals and sums, 4th power moment, zero-free region, mean value estimates over short intervals, higher power moments, omega results, zeros on the critical line, zero-density estimates, and more. 1985 edition.
| Author | : S. Barry Cooper |
| Publisher | : Elsevier |
| Total Pages | : 937 |
| Release | : 2013-03-18 |
| Genre | : Mathematics |
| ISBN | : 0123870127 |
In this 2013 winner of the prestigious R.R. Hawkins Award from the Association of American Publishers, as well as the 2013 PROSE Awards for Mathematics and Best in Physical Sciences & Mathematics, also from the AAP, readers will find many of the most significant contributions from the four-volume set of the Collected Works of A. M. Turing. These contributions, together with commentaries from current experts in a wide spectrum of fields and backgrounds, provide insight on the significance and contemporary impact of Alan Turing's work. Offering a more modern perspective than anything currently available, Alan Turing: His Work and Impact gives wide coverage of the many ways in which Turing's scientific endeavors have impacted current research and understanding of the world. His pivotal writings on subjects including computing, artificial intelligence, cryptography, morphogenesis, and more display continued relevance and insight into today's scientific and technological landscape. This collection provides a great service to researchers, but is also an approachable entry point for readers with limited training in the science, but an urge to learn more about the details of Turing's work. - 2013 winner of the prestigious R.R. Hawkins Award from the Association of American Publishers, as well as the 2013 PROSE Awards for Mathematics and Best in Physical Sciences & Mathematics, also from the AAP - Named a 2013 Notable Computer Book in Computing Milieux by Computing Reviews - Affordable, key collection of the most significant papers by A.M. Turing - Commentary explaining the significance of each seminal paper by preeminent leaders in the field - Additional resources available online
| Author | : Michel Laurent Lapidus |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 594 |
| Release | : 2008 |
| Genre | : Mathematics |
| ISBN | : 9780821842225 |
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.
| Author | : Audrey Terras |
| Publisher | : Cambridge University Press |
| Total Pages | : 253 |
| Release | : 2010-11-18 |
| Genre | : Mathematics |
| ISBN | : 1139491784 |
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.
| Author | : André Voros |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 171 |
| Release | : 2009-11-21 |
| Genre | : Mathematics |
| ISBN | : 3642052037 |
In this text, the famous zeros of the Riemann zeta function and its generalizations (L-functions, Dedekind and Selberg zeta functions)are analyzed through several zeta functions built over those zeros.
| Author | : Michel L. Lapidus |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 583 |
| Release | : 2012-09-20 |
| Genre | : Mathematics |
| ISBN | : 1461421764 |
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.
