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Author: Tom M. Apostol
Publisher: Springer Science & Business Media
ISBN: 1461209994
Category : Mathematics
Languages : en
Pages : 218
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Book Description
A new edition of a classical treatment of elliptic and modular functions with some of their number-theoretic applications, this text offers an updated bibliography and an alternative treatment of the transformation formula for the Dedekind eta function. It covers many topics, such as Hecke’s theory of entire forms with multiplicative Fourier coefficients, and the last chapter recounts Bohr’s theory of equivalence of general Dirichlet series.
Author: Tom M. Apostol
Publisher: Springer Science & Business Media
ISBN: 1461209994
Category : Mathematics
Languages : en
Pages : 218
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Book Description
A new edition of a classical treatment of elliptic and modular functions with some of their number-theoretic applications, this text offers an updated bibliography and an alternative treatment of the transformation formula for the Dedekind eta function. It covers many topics, such as Hecke’s theory of entire forms with multiplicative Fourier coefficients, and the last chapter recounts Bohr’s theory of equivalence of general Dirichlet series.
Author: Goro Shimura
Publisher: Springer Science & Business Media
ISBN: 0387724745
Category : Mathematics
Languages : en
Pages : 152
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Book Description
A book on any mathematical subject beyond the textbook level is of little value unless it contains new ideas and new perspectives. It helps to include new results, provided that they give the reader new insights and are presented along with known old results in a clear exposition. It is with this philosophy that the author writes this volume. The two subjects, Dirichlet series and modular forms, are traditional subjects, but here they are treated in both orthodox and unorthodox ways. Regardless of the unorthodox treatment, the author has made the book accessible to those who are not familiar with such topics by including plenty of expository material.
Author: Anatoli Andrianov
Publisher: Springer Science & Business Media
ISBN: 0387787534
Category : Mathematics
Languages : en
Pages : 188
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Book Description
Several years ago I was invited to an American university to give one-term graduate course on Siegel modular forms, Hecke operators, and related zeta functions. The idea to present in a concise but basically complete and self-contained form an int- duction to an important and developing area based partly on my own work attracted me. I accepted the invitation and started to prepare the course. Unfortunately, the visit was not realized. But the idea of such a course continued to be alive till after a number of years this book was ?nally completed. I hope that this short book will serve to attract young researchers to this beautiful ?eld, and that it will simplify and make more pleasant the initial steps. No special knowledge is presupposed for reading this book beyond standard courses in algebra and calculus (one and several variables), although some skill in working with mathematical texts would be helpful. The reader will judge whether the result was worth the effort. Dedications. The ideas of Goro Shimura exerted a deep in?uence on the number theory of the second half of the twentieth century in general and on the author’s formation in particular. When Andre ` Weil was signing a copy of his “Basic Number Theory” to my son, he wrote in Russian, ”To Fedor Anatolievich hoping that he will become a number theoretist”. Fedor has chosen computer science. Now I pass on the idea to Fedor’s daughter, Alexandra Fedorovna.
Author: Bruce C. Berndt
Publisher: World Scientific
ISBN: 9812706356
Category : Mathematics
Languages : en
Pages : 150
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Book Description
Cyber security, encompassing both information and network security, is of utmost importance in today's information age. Cyber Security Standards, Practices and Industrial Applications: Systems and Methodologies details the latest and most important advances in security standards. First, it introduces the differences between information security (covers the understanding of security requirements, classification of threats, attacks and information protection systems and methodologies) and network security (includes both security protocols as well as systems which create a security perimeter around networks for intrusion detection and avoidance). In addition, the book serves as an essential reference to students, researchers, practitioners, and consultants in the area of social media, cyber security and information, and communication technologies (ICT).
Author: Henri Cohen
Publisher: American Mathematical Soc.
ISBN: 0821849476
Category : Forms (Mathematics).
Languages : en
Pages : 700
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Book Description
The theory of modular forms is a fundamental tool used in many areas of mathematics and physics. It is also a very concrete and “fun” subject in itself and abounds with an amazing number of surprising identities. This comprehensive textbook, which includes numerous exercises, aims to give a complete picture of the classical aspects of the subject, with an emphasis on explicit formulas. After a number of motivating examples such as elliptic functions and theta functions, the modular group, its subgroups, and general aspects of holomorphic and nonholomorphic modular forms are explained, with an emphasis on explicit examples. The heart of the book is the classical theory developed by Hecke and continued up to the Atkin–Lehner–Li theory of newforms and including the theory of Eisenstein series, Rankin–Selberg theory, and a more general theory of theta series including the Weil representation. The final chapter explores in some detail more general types of modular forms such as half-integral weight, Hilbert, Jacobi, Maass, and Siegel modular forms. Some “gems” of the book are an immediately implementable trace formula for Hecke operators, generalizations of Haberland's formulas for the computation of Petersson inner products, W. Li's little-known theorem on the diagonalization of the full space of modular forms, and explicit algorithms due to the second author for computing Maass forms. This book is essentially self-contained, the necessary tools such as gamma and Bessel functions, Bernoulli numbers, and so on being given in a separate chapter.
Author: Goro Shimura
Publisher: Springer Science & Business Media
ISBN: 146142125X
Category : Mathematics
Languages : en
Pages : 178
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Book Description
This is an advanced book on modular forms. While there are many books published about modular forms, they are written at an elementary level, and not so interesting from the viewpoint of a reader who already knows the basics. This book offers something new, which may satisfy the desire of such a reader. However, we state every definition and every essential fact concerning classical modular forms of one variable. One of the principal new features of this book is the theory of modular forms of half-integral weight, another being the discussion of theta functions and Eisenstein series of holomorphic and nonholomorphic types. Thus the book is presented so that the reader can learn such theories systematically.
Author: Tom M. Apostol
Publisher: Springer Science & Business Media
ISBN: 1468499106
Category : Mathematics
Languages : en
Pages : 207
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Book Description
This is the second volume of a 2-volume textbook* which evolved from a course (Mathematics 160) offered at the California Institute of Technology du ring the last 25 years. The second volume presupposes a background in number theory com parable to that provided in the first volume, together with a knowledge of the basic concepts of complex analysis. Most of the present volume is devoted to elliptic functions and modular functions with some of their number-theoretic applications. Among the major topics treated are Rademacher's convergent series for the partition function, Lehner's congruences for the Fourier coefficients of the modular functionj( r), and Hecke's theory of entire forms with multiplicative Fourier coefficients. The last chapter gives an account of Bohr's theory of equivalence of general Dirichlet series. Both volumes of this work emphasize classical aspects of a subject wh ich in recent years has undergone a great deal of modern development. It is hoped that these volumes will help the nonspecialist become acquainted with an important and fascinating part of mathematics and, at the same time, will provide some of the background that belongs to the repertory of every specialist in the field. This volume, like the first, is dedicated to the students who have taken this course and have gone on to make notable contributions to number theory and other parts of mathematics. T. M. A. January, 1976 * The first volume is in the Springer-Verlag series Undergraduate Texts in Mathematics under the title Introduction to Analytic Number Theory.
Author: Andrew Ogg
Publisher:
ISBN: 9780805375756
Category :
Languages : en
Pages : 16
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Book Description
Author: Hans Maass
Publisher: Springer
ISBN:
Category : Mathematics
Languages : en
Pages : 348
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Book Description
These notes present the content of a course delivered at the University of Maryland, College Park, between September 1969 and April 1970. The subject is mainly by the intention to show how Atle Selberg makes fascinating use of differential operators in order to prove certain functional equations.
Author: L J P Kilford
Publisher: World Scientific
ISBN: 190897883X
Category : Mathematics
Languages : en
Pages : 236
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Book Description
This book presents a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to such diverse subjects as the theory of quadratic forms, the proof of Fermat's last theorem and the approximation of pi. It provides a balanced overview of both the theoretical and computational sides of the subject, allowing a variety of courses to be taught from it. Contents: Historical OverviewIntroduction to Modular FormsResults on Finite-DimensionalityThe Arithmetic of Modular FormsApplications of Modular FormsModular Forms in Characteristic pComputing with Modular FormsAppendices:MAGMA Code for Classical Modular FormsSAGE Code for Classical Modular FormsHints and Answers to Selected Exercises Readership: Academics, researchers and graduate students in number theory and computational mathematics. Keywords:Modular Forms;Computations;Modular Functions;Cusp Forms;Ramanujan Tau FunctionKey Features:Covers the computational side together with the theoryIncludes a wide variety of exercises, from short to research-project lengthContains historical asides and references to modular forms in mathematical culture, to help ground the subject and motivate student interestReviews: "This fascinating, contemporaneous, and even now unfolding story of current research in a historically brilliant part of mathematics is told with riveting attention to detail ... Almost all aspects one could wish for in the area of holomorphic modular forms are covered, as well as some selected topics about meromorphic modular functions." The Mathematical Intelligencer "The second and (perhaps) more interesting computational aspect conveyed in this book is the consistent use of explicit computations by hand. For example expressing modular forms in a given space in terms of Eisenstein series, Eta or Delta functions to verify and prove various statements and theorems. This aspect is further encouraged throughout the exercises, which by the way are numerous, relevant and well-written. This kind of very explicit computations are sadly missing in the literature although implicitly stated or used in many places. It is obviously well-known to experts but most students would never be exposed to these ideas unless actually playing around to prove theorems by themselves." Zentrallblatt MATH
