Congruences for L-Functions

By: J. Urbanowicz, Kenneth S. Williams

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Hardcover | 30 June 2000

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276 Pages

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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 idl/8,="" gcd(k,="" d)="1," gives="" ~="" (-it(e)="" ~="" (~)=":O(mod2n)." eld="">

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Preface
Short Character Sums	p. 1
Introduction	p. 1
Bernoulli Numbers and Polynomials	p. 9
Generalized Bernoulli Numbers	p. 12
Dirichlet L-functions	p. 16
The values of L(1,[chi])and L(2,[chi])	p. 18
The Dedekind Zeta Function	p. 21
K-theoretic Background	p. 23
Quadratic Fields	p. 27
Power Sums Involving Dirichlet Characters	p. 30
Some Elementary Lemmas	p. 45
Class Number Congruences	p. 51
Imaginary Quadratic Fields	p. 51
Real Quadratic Fields	p. 72
Congruences Between the Orders of K[subscript 2]-Groups	p. 77
Real Quadratic Fields	p. 78
Congruences for Higher Bernoulli Numbers	p. 96
Congruences Among the Values of 2-adic L-functions	p. 117
Notation	p. 117
p-adic L-functions	p. 119
Coleman's Results	p. 122
Some Auxiliary Lemmas	p. 128
Linear Congruence Relations	p. 160
Applications of Zagier's Formula (I)	p. 181
Introduction	p. 181
Dirichlet Characters with Certain Properties	p. 184
Character Sums in Terms of Bernoulli Numbers	p. 192
Applications	p. 195
Tables	p. 198
Applications of Zagier's Formula (II)	p. 203
Preliminaries	p. 203
Gauss' Congruence from Dirichlet's Class Number Formula	p. 208
Character Power Sums in Terms of Bernoulli Numbers	p. 210
The Main Results	p. 213
Bibliography	p. 231
Author Index	p. 247
Subject Index	p. 249
List of symbols	p. 253
Table of Contents provided by Blackwell. All Rights Reserved.

Congruences for L-Functions

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