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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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Bibliographic Information
- Bibliographic Information
-
- Book Title
- Congruences for L-Functions
- Authors
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- J. Urbanowicz
- Kenneth S. Williams
- Series Title
- Mathematics and Its Applications
- Series Volume
- 511
- Copyright
- 2000
- Publisher
- Springer Netherlands
- Copyright Holder
- Springer Science+Business Media B.V.
- eBook ISBN
- 978-94-015-9542-1
- DOI
- 10.1007/978-94-015-9542-1
- Hardcover ISBN
- 978-0-7923-6379-8
- Softcover ISBN
- 978-90-481-5490-6
- Edition Number
- 1
- Number of Pages
- XII, 256
- Topics
