Mathematics and Its Applications

# Congruences for L-Functions

Authors: Urbanowicz, J., Williams, Kenneth S.

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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

eBook 79,72 €
price for Spain (gross)
valid through March 31, 2018
• ISBN 978-94-015-9542-1
• Digitally watermarked, DRM-free
• Included format: PDF
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Hardcover 98,79 €
price for Spain (gross)
valid through March 31, 2018
• ISBN 978-0-7923-6379-8
• Free shipping for individuals worldwide
• Usually dispatched within 3 to 5 business days.
• The final prices may differ from the prices shown due to specifics of VAT rules
Softcover 103,99 €
price for Spain (gross)
• ISBN 978-90-481-5490-6
• Free shipping for individuals worldwide
• Usually dispatched within 3 to 5 business days.
• The final prices may differ from the prices shown due to specifics of VAT rules

## Bibliographic Information

Bibliographic Information
Book Title
Congruences for L-Functions
Authors
Series Title
Mathematics and Its Applications
Series Volume
511
2000
Publisher
Springer Netherlands