Ebook: Non-vanishing of L-Functions and Applications

This book systematically develops some methods for proving the non-vanishing of certain L-functions at points in the critical strip. Researchers in number theory, graduate students who wish to enter into the area and non-specialists who wish to acquire an introduction to the subject will benefit by a study of this book. One of the most attractive features of the monograph is that it begins at a very basic level and quickly develops enough aspects of the theory to bring the reader to a point where the latest discoveries as are presented in the final chapters can be fully appreciated.

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This book has been awarded the Ferran Sunyer I Balaguer 1996 prize (…)The deepest results are contained in Chapter 6 on quadratic twists of modular L-functions with connections to the Birch-Swinnerton-Dyer conjecture. (…) [It] is well-suited and stimulating for the graduate level because there is a wealth of recent results and open problems, and also a number of exercices and references after each chapter.

(Zentralblatt MATH)

Each chapter is accompanied by exercices, and there is a fair amount of introductory material, general discussion and recommended reading. (…) it will be a useful addition to the library of any serious worker in this area.

(Mathematical Reviews)

(…) well written monograph, intended not only for researchers and graduate students specializing in number theory, but also for non-specialists desiring to acquire an introduction to this difficult but very attractive and beautiful domain of investigation.

(Mathematica)

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The theory of finite fields is of central importance in engineering and computer science, because of its applications to error-correcting codes, cryptography, spread-spectrum communications, and digital signal processing. Though not inherently difficult, this subject is almost never taught in depth in mathematics courses, (and even when it is the emphasis is rarely on the practical aspect). Indeed, most students get a brief and superficial survey which is crammed into a course on error-correcting codes. It is the object of this text to remedy this situation by presenting a thorough introduction to the subject which is completely sound mathematically, yet emphasizes those aspects of the subject which have proved to be the most important for applications. This book is unique in several respects. Throughout, the emphasis is on fields of characteristic 2, the fields on which almost all applications are based. The importance of Euclid's algorithm is stressed early and often. Berlekamp's polynomial factoring algorithm is given a complete explanation. The book contains the first treatment of Berlekamp's 1982 bit-serial multiplication circuits, and concludes with a thorough discussion of the theory of m-sequences, which are widely used in communications systems of many kinds The prime number theorem and generalizations -- Artin L-functions -- Equidistribution and L-functions -- Modular forms and Dirichlet series -- Dirichlet L-functions -- Non-vanishing of quadratic twists of modular L-functions -- Selberg's conjectures -- Suggestions for further reading