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＜E-Book＞
Congruences for L-Functions / by J. Urbanowicz, Kenneth S. Williams
(Mathematics and Its Applications ; 511)

Edition	1st ed. 2000.
Publisher	Dordrecht : Springer Netherlands : Imprint: Springer
Year	2000
Language	English
Size	XII, 256 p : online resource
Authors	*Urbanowicz, J author Williams, Kenneth S author SpringerLink (Online service)
Subjects	LCSH:Number theory LCSH:Algebraic fields LCSH:Polynomials LCSH:Functions of complex variables LCSH:Special functions FREE:Number Theory FREE:Field Theory and Polynomials FREE:Functions of a Complex Variable FREE:Special Functions
Notes	I. Short Character Sums -- II. Class Number Congruences -- III. Congruences between the Orders of K2-Groups -- IV Congruences among the Values of 2-Adic L-Functions -- V. Applications of Zagier’s Formula (I) -- VI. Applications of Zagier’s Formula (II) -- Author Index -- List of symbols 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 HTTP:URL=https://doi.org/10.1007/978-94-015-9542-1

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