From Arithmetic To Zeta Functions
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Author |
: Spencer J. Bloch |
Publisher |
: American Mathematical Soc. |
Total Pages |
: 114 |
Release |
: 2011 |
ISBN-10 |
: 9780821829738 |
ISBN-13 |
: 0821829734 |
Rating |
: 4/5 (38 Downloads) |
Synopsis Higher Regulators, Algebraic $K$-Theory, and Zeta Functions of Elliptic Curves by : Spencer J. Bloch
This is the long-awaited publication of the famous Irvine lectures. Delivered in 1978 at the University of California at Irvine, these lectures turned out to be an entry point to several intimately-connected new branches of arithmetic algebraic geometry, such as regulators and special values of L-functions of algebraic varieties, explicit formulas for them in terms of polylogarithms, the theory of algebraic cycles, and eventually the general theory of mixed motives which unifies and underlies all of the above (and much more).
Author |
: Jürgen Sander |
Publisher |
: Springer |
Total Pages |
: 552 |
Release |
: 2016-12-29 |
ISBN-10 |
: 9783319282039 |
ISBN-13 |
: 3319282034 |
Rating |
: 4/5 (39 Downloads) |
Synopsis From Arithmetic to Zeta-Functions by : Jürgen Sander
This book collects more than thirty contributions in memory of Wolfgang Schwarz, most of which were presented at the seventh International Conference on Elementary and Analytic Number Theory (ELAZ), held July 2014 in Hildesheim, Germany. Ranging from the theory of arithmetical functions to diophantine problems, to analytic aspects of zeta-functions, the various research and survey articles cover the broad interests of the well-known number theorist and cherished colleague Wolfgang Schwarz (1934-2013), who contributed over one hundred articles on number theory, its history and related fields. Readers interested in elementary or analytic number theory and related fields will certainly find many fascinating topical results among the contributions from both respected mathematicians and up-and-coming young researchers. In addition, some biographical articles highlight the life and mathematical works of Wolfgang Schwarz.
Author |
: H. Iwaniec |
Publisher |
: American Mathematical Society |
Total Pages |
: 130 |
Release |
: 2014-10-07 |
ISBN-10 |
: 9781470418519 |
ISBN-13 |
: 1470418517 |
Rating |
: 4/5 (19 Downloads) |
Synopsis Lectures on the Riemann Zeta Function by : H. Iwaniec
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 |
: Gorō Shimura |
Publisher |
: Princeton University Press |
Total Pages |
: 292 |
Release |
: 1971-08-21 |
ISBN-10 |
: 0691080925 |
ISBN-13 |
: 9780691080925 |
Rating |
: 4/5 (25 Downloads) |
Synopsis Introduction to the Arithmetic Theory of Automorphic Functions by : Gorō Shimura
The theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and in particular to elliptic modular forms with emphasis on their number-theoretical aspects. After two chapters geared toward elementary levels, there follows a detailed treatment of the theory of Hecke operators, which associate zeta functions to modular forms. At a more advanced level, complex multiplication of elliptic curves and abelian varieties is discussed. The main question is the construction of abelian extensions of certain algebraic number fields, which is traditionally called "Hilbert's twelfth problem." Another advanced topic is the determination of the zeta function of an algebraic curve uniformized by modular functions, which supplies an indispensable background for the recent proof of Fermat's last theorem by Wiles.
Author |
: John Coates |
Publisher |
: Cambridge University Press |
Total Pages |
: 317 |
Release |
: 2015-03-19 |
ISBN-10 |
: 9781316241301 |
ISBN-13 |
: 1316241300 |
Rating |
: 4/5 (01 Downloads) |
Synopsis The Bloch–Kato Conjecture for the Riemann Zeta Function by : John Coates
There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch–Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings.
Author |
: Dinesh S. Thakur |
Publisher |
: World Scientific |
Total Pages |
: 405 |
Release |
: 2004 |
ISBN-10 |
: 9789812388391 |
ISBN-13 |
: 9812388397 |
Rating |
: 4/5 (91 Downloads) |
Synopsis Function Field Arithmetic by : Dinesh S. Thakur
This book provides an exposition of function field arithmetic with emphasis on recent developments concerning Drinfeld modules, the arithmetic of special values of transcendental functions (such as zeta and gamma functions and their interpolations), diophantine approximation and related interesting open problems. While it covers many topics treated in 'Basic Structures of Function Field Arithmetic' by David Goss, it complements that book with the inclusion of recent developments as well as the treatment of new topics such as diophantine approximation, hypergeometric functions, modular forms, transcendence, automata and solitons. There is also new work on multizeta values and log-algebraicity. The author has included numerous worked-out examples. Many open problems, which can serve as good thesis problems, are discussed.
Author |
: John Voight |
Publisher |
: Springer Nature |
Total Pages |
: 877 |
Release |
: 2021-06-28 |
ISBN-10 |
: 9783030566944 |
ISBN-13 |
: 3030566943 |
Rating |
: 4/5 (44 Downloads) |
Synopsis Quaternion Algebras by : John Voight
This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout.
Author |
: Antanas Laurincikas |
Publisher |
: Springer Science & Business Media |
Total Pages |
: 192 |
Release |
: 2013-12-11 |
ISBN-10 |
: 9789401764018 |
ISBN-13 |
: 9401764018 |
Rating |
: 4/5 (18 Downloads) |
Synopsis The Lerch zeta-function by : Antanas Laurincikas
The Lerch zeta-function is the first monograph on this topic, which is a generalization of the classic Riemann, and Hurwitz zeta-functions. Although analytic results have been presented previously in various monographs on zeta-functions, this is the first book containing both analytic and probability theory of Lerch zeta-functions. The book starts with classical analytical theory (Euler gamma-functions, functional equation, mean square). The majority of the presented results are new: on approximate functional equations and its applications and on zero distribution (zero-free regions, number of nontrivial zeros etc). Special attention is given to limit theorems in the sense of the weak convergence of probability measures for the Lerch zeta-function. From limit theorems in the space of analytic functions the universitality and functional independence is derived. In this respect the book continues the research of the first author presented in the monograph Limit Theorems for the Riemann zeta-function. This book will be useful to researchers and graduate students working in analytic and probabilistic number theory, and can also be used as a textbook for postgraduate students.
Author |
: Anatoly A. Karatsuba |
Publisher |
: Walter de Gruyter |
Total Pages |
: 409 |
Release |
: 2011-05-03 |
ISBN-10 |
: 9783110886146 |
ISBN-13 |
: 3110886146 |
Rating |
: 4/5 (46 Downloads) |
Synopsis The Riemann Zeta-Function by : Anatoly A. Karatsuba
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 |
: Jianqiang Zhao |
Publisher |
: World Scientific |
Total Pages |
: 618 |
Release |
: 2016-03-07 |
ISBN-10 |
: 9789814689410 |
ISBN-13 |
: 9814689416 |
Rating |
: 4/5 (10 Downloads) |
Synopsis Multiple Zeta Functions, Multiple Polylogarithms And Their Special Values by : Jianqiang Zhao
This is the first introductory book on multiple zeta functions and multiple polylogarithms which are the generalizations of the Riemann zeta function and the classical polylogarithms, respectively, to the multiple variable setting. It contains all the basic concepts and the important properties of these functions and their special values. This book is aimed at graduate students, mathematicians and physicists who are interested in this current active area of research.The book will provide a detailed and comprehensive introduction to these objects, their fascinating properties and interesting relations to other mathematical subjects, and various generalizations such as their q-analogs and their finite versions (by taking partial sums modulo suitable prime powers). Historical notes and exercises are provided at the end of each chapter.