Congruences for L-Functions

1. Front Matter

   Pages i-xii

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2. Short Character Sums

   • Jerzy Urbanowicz, Kenneth S. Williams

   Pages 1-49

3. Class Number Congruences

   • Jerzy Urbanowicz, Kenneth S. Williams

   Pages 51-76

4. Congruences Between the Orders of K ₂-Groups

   • Jerzy Urbanowicz, Kenneth S. Williams

   Pages 77-116

5. Congruences among the Values of 2-Adic L-Functions

   • Jerzy Urbanowicz, Kenneth S. Williams

   Pages 117-180

6. Applications of Zagier’s Formula (I)

   • Jerzy Urbanowicz, Kenneth S. Williams

   Pages 181-202

7. Applications of Zagier’s Formula (II)

   • Jerzy Urbanowicz, Kenneth S. Williams

   Pages 203-230

8. Back Matter

   Pages 231-256

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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

• DEX
• character
• congruence
• form
• function
• functions
• number theory
• special function
• variable

• Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland

  Jerzy Urbanowicz

• Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Canada

  Kenneth S. Williams

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