Congruences for L-Functions

Jerzy Urbanowicz, Kenneth S. Williams (auth.)0792363795, 9780792363798

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

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Table of contents :
Front Matter….Pages i-xii
Short Character Sums….Pages 1-49
Class Number Congruences….Pages 51-76
Congruences Between the Orders of K 2 -Groups….Pages 77-116
Congruences among the Values of 2-Adic L -Functions….Pages 117-180
Applications of Zagier’s Formula (I)….Pages 181-202
Applications of Zagier’s Formula (II)….Pages 203-230
Back Matter….Pages 231-256