Elementary Dirichlet Series and Modular Forms / Edition 1

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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.
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Editorial Reviews

From the Publisher

From the reviews:

"This book contains new results, e.g., new formulas for special values of certain Dirichlet series. … Shimura’s exposition, shaped to his (celebrated and) distinctive viewpoint, serves as the ideal platform for the new material. … by dint of the prestige of the author and the subject, it undoubtedly deserves a place in a college library. … Summing Up: Recommended. Upper-division undergraduates through faculty." (D. V. Feldman, CHOICE, Vol. 45 (9), 2008)

"It will be of great interest for everybody who is interested in modular forms and/or L-series. … the monograph will be accessible to graduate students and will quickly lead them to frontiers of current research. The book is written in the well-known masterly style of the author … ." (Jürgen Elstrodt, Zentralblatt MATH, Vol. 1148, 2008)

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Product Details

  • ISBN-13: 9780387724737
  • Publisher: Springer New York
  • Publication date: 9/28/2007
  • Series: Springer Monographs in Mathematics Series
  • Edition description: 2007
  • Edition number: 1
  • Pages: 152
  • Product dimensions: 6.20 (w) x 9.30 (h) x 0.50 (d)

Table of Contents

Preface     V
Introduction     1
Preliminaries on Modular Forms and Dirichlet Series     5
Basic symbols and the definition of modular forms     5
Elementary Fourier analysis     13
The functional equation of a Dirichlet series     19
Critical Values of Dirichlet L-functions     25
The values of elementary Dirichlet series at integers     25
The class number of a cyclotomic field     39
Some more formulas for L(k, x)     45
The Case of Imaginary Quadratic Fields and Nearly Holomorphic Modular Forms     53
Dirichlet series associated with an imaginary quadratic field     53
Nearly holomorphic modular forms     55
Eisenstein Series     59
Fourier expansion of Eisenstein series     59
Polynomial relations between Eisenstein series     66
Recurrence formulas for the critical values of certain Dirichlet series     75
Critical Values of Dirichlet Series Associated with Imaginary Quadratic Fields     79
The singular values of nearly holomorphic forms     79
The critical values of L-functions of an imaginary quadratic field     84
The zeta function of a member of a one-parameter family of elliptic curves     96
Supplementary Results     113
Isomorphism classes of abelian varieties with complex multiplication     113
The general case     113
The case of elliptic curves     117
Holomorphic differential operators on the upper half plane     120
Appendix     127
Integration and differentiation under the integral sign     127
Fourier series with parameters     130
The confluent hypergeometric function     131
The Weierstrass [Characters not reproducible]-function     136
The action of G[subscript A+] on modular forms     139
References     145
Index     147
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