# Congruences for L-Functions by Jerzy Urbanowicz, Kenneth S. Williams (auth.)

By Jerzy Urbanowicz, Kenneth S. Williams (auth.)

In [Hardy and Williams, 1986] the authors exploited a very easy concept to procure a linear congruence related to category numbers of imaginary quadratic fields modulo a definite energy of two. Their congruence supplied a unified environment for plenty of congruences proved formerly via different authors utilizing a variety of potential. The Hardy-Williams concept was once as follows. allow d be the discriminant of a quadratic box. consider that d is peculiar and enable d = PIP2· . . Pn be its distinctive decomposition into best discriminants. Then, for any optimistic integer okay coprime with d, the congruence holds trivially as every one Legendre-Jacobi-Kronecker image (~) has the worth + 1 or -1. increasing this product supplies ~ eld e:=l (mod4) the place e runs throughout the optimistic and damaging divisors of d and v (e) denotes the variety of particular best elements of e. Summing this congruence for o < okay < Idl/8, gcd(k, d) = 1, provides ~ (-It(e) ~ (~) =:O(mod2n). eld o

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5 ([Szmidt and Urbanowicz, 1994]) Let X be a primitive Dirichlet character with odd conductor M and let N ~ 1 be an odd multiple of M. Thenfor m ~ 0, we have THEOREM 23m +2 (m + I)Sm(N/8, X) = -X_8(N)x(8)Bm+1 ,x_8X(N) - X_4(N)x(8)Bm+1 ,L4X(N) + X8(N)x(8)Bm+ 1 ,X8 x (N) + x(8)Bm+1 ,x(N) - 2mx(4)Bm+l,x(N/2) - 23m+2 Bm+l,x' Theorem 5 follows from either Theorem 1 or Theorem 2. In [Szmidt and Urbanowicz, 1994] the authors used the formula of Theorem 1 to prove Theorem 5. Their proof is considerably longer than the proof using Theorem 2, which we leave to the reader.

21.. (see [Tate, 1973], [Skalba, 1987, 1994] and [Qin, 1994b, 1996]). /21.. for D = -23, -31. Belabas and Gangl ([Belabas and Gangl, 1999]) have determined the group K 2 0 F for all negative discriminants down to -151. For real quadratic fields F much more is known. The groups K 2 0 F are determined up to several thousand. This is due to the fact that the orders of the groups K 2 0 F can be expressed by means of corresponding Bernoulli numbers and so are easily computed. See also [Conner and Hurrelbrink, 1986], [Qin, 1994a, 1995a,b, 1998], [Vazzana, 1997a,b, 1999] and [Browkin and Gangl, 1999].

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