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Congruences for L-Functions

  • Book
  • © 2000

Overview

Authors:
  1. Jerzy Urbanowicz

    1. Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland

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  2. Kenneth S. Williams

    1. Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Canada

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    You can also search for this author in PubMed   Google Scholar

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

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xii
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  2. Short Character Sums

    • Jerzy Urbanowicz, Kenneth S. Williams
    Pages 1-49

  3. Class Number Congruences

    • Jerzy Urbanowicz, Kenneth S. Williams
    Pages 51-76

  4. Congruences Between the Orders of K 2-Groups

    • Jerzy Urbanowicz, Kenneth S. Williams
    Pages 77-116

  5. Congruences among the Values of 2-Adic L-Functions

    • Jerzy Urbanowicz, Kenneth S. Williams
    Pages 117-180

  6. Applications of Zagier’s Formula (I)

    • Jerzy Urbanowicz, Kenneth S. Williams
    Pages 181-202

  7. Applications of Zagier’s Formula (II)

    • Jerzy Urbanowicz, Kenneth S. Williams
    Pages 203-230

  8. Back Matter

    Pages 231-256
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Keywords

About this book

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

Authors and Affiliations

  • Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland

    Jerzy Urbanowicz

  • Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Canada

    Kenneth S. Williams

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