Congruences for L-Functions

  • Jerzy Urbanowicz
  • Kenneth S. Williams

Part of the Mathematics and Its Applications book series (MAIA, volume 511)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Jerzy Urbanowicz, Kenneth S. Williams
    Pages 1-49
  3. Jerzy Urbanowicz, Kenneth S. Williams
    Pages 51-76
  4. Jerzy Urbanowicz, Kenneth S. Williams
    Pages 77-116
  5. Jerzy Urbanowicz, Kenneth S. Williams
    Pages 117-180
  6. Jerzy Urbanowicz, Kenneth S. Williams
    Pages 181-202
  7. Jerzy Urbanowicz, Kenneth S. Williams
    Pages 203-230
  8. Back Matter
    Pages 231-256

About this book

Introduction

In [Hardy and Williams, 1986] the authors exploited a very simple idea to obtain a linear congruence involving class numbers of imaginary quadratic fields modulo a certain power of 2. Their congruence provided a unified setting for many congruences proved previously by other authors using various means. The Hardy-Williams idea was as follows. Let d be the discriminant of a quadratic field. Suppose that d is odd and let d = PIP2· . . Pn be its unique decomposition into prime discriminants. Then, for any positive integer k coprime with d, the congruence holds trivially as each Legendre-Jacobi-Kronecker symbol (~) has the value + 1 or -1. Expanding this product gives ~ eld e:=l (mod4) where e runs through the positive and negative divisors of d and v (e) denotes the number of distinct prime factors of e. Summing this congruence for o < k < Idl/8, gcd(k, d) = 1, gives ~ (-It(e) ~ (~) =:O(mod2n). eld o

Keywords

DEX character congruence form function functions number theory special function variable

Authors and affiliations

  • Jerzy Urbanowicz
    • 1
  • Kenneth S. Williams
    • 2
  1. 1.Institute of MathematicsPolish Academy of SciencesWarszawaPoland
  2. 2.Centre for Research in Algebra and Number Theory, School of Mathematics and StatisticsCarleton UniversityOttawaCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-015-9542-1
  • Copyright Information Springer Science+Business Media B.V. 2000
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-5490-6
  • Online ISBN 978-94-015-9542-1
  • About this book
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