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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
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Table of contents (6 chapters)
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Bibliographic Information
Book Title: Congruences for L-Functions
Authors: Jerzy Urbanowicz, Kenneth S. Williams
Series Title: Mathematics and Its Applications
DOI: https://doi.org/10.1007/978-94-015-9542-1
Publisher: Springer Dordrecht
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eBook Packages: Springer Book Archive
Copyright Information: Springer Science+Business Media B.V. 2000
Hardcover ISBN: 978-0-7923-6379-8Published: 30 June 2000
Softcover ISBN: 978-90-481-5490-6Published: 15 December 2010
eBook ISBN: 978-94-015-9542-1Published: 09 March 2013
Edition Number: 1
Number of Pages: XII, 256
Topics: Number Theory, Field Theory and Polynomials, Functions of a Complex Variable, Special Functions