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© 2000

Congruences for L-Functions

Book

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

  1. 1.Institute of MathematicsPolish Academy of SciencesWarszawaPoland
  2. 2.Centre for Research in Algebra and Number Theory, School of Mathematics and StatisticsCarleton UniversityOttawaCanada

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