Congruences for L-Functions

Congruences for L-Functions

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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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Product details

  • Hardback | 256 pages
  • 156 x 234 x 17.53mm | 1,250g
  • Dordrecht, Netherlands
  • English
  • 2000 ed.
  • XII, 256 p.
  • 0792363795
  • 9780792363798

Table of contents

Preface. 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). Bibliography. Author Index. Subject Index. List of symbols.
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