Ebook: Congruences for L-Functions

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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This book provides a comprehensive and up-to-date treatment of research carried out in the last twenty years on congruences involving the values of L-functions (attached to quadratic characters) at certain special values.
There is no other book on the market which deals with this subject. The book presents in a unified way congruences found by many authors over the years, from the classical ones of Gauss and Dirichlet to the recent ones of Gras, Vehara, and others.
Audience: This book is aimed at graduate students and researchers interested in (analytic) number theory, functions of a complex variable and special functions.

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This book provides a comprehensive and up-to-date treatment of research carried out in the last twenty years on congruences involving the values of L-functions (attached to quadratic characters) at certain special values.
There is no other book on the market which deals with this subject. The book presents in a unified way congruences found by many authors over the years, from the classical ones of Gauss and Dirichlet to the recent ones of Gras, Vehara, and others.
Audience: This book is aimed at graduate students and researchers interested in (analytic) number theory, functions of a complex variable and special functions.
Content:
Front Matter....Pages i-xii
Short Character Sums....Pages 1-49
Class Number Congruences....Pages 51-76
Congruences Between the Orders of K 2-Groups....Pages 77-116
Congruences among the Values of 2-Adic L-Functions....Pages 117-180
Applications of Zagier’s Formula (I)....Pages 181-202
Applications of Zagier’s Formula (II)....Pages 203-230
Back Matter....Pages 231-256

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This book provides a comprehensive and up-to-date treatment of research carried out in the last twenty years on congruences involving the values of L-functions (attached to quadratic characters) at certain special values.
There is no other book on the market which deals with this subject. The book presents in a unified way congruences found by many authors over the years, from the classical ones of Gauss and Dirichlet to the recent ones of Gras, Vehara, and others.
Audience: This book is aimed at graduate students and researchers interested in (analytic) number theory, functions of a complex variable and special functions.
Content:
Front Matter....Pages i-xii
Short Character Sums....Pages 1-49
Class Number Congruences....Pages 51-76
Congruences Between the Orders of K 2-Groups....Pages 77-116
Congruences among the Values of 2-Adic L-Functions....Pages 117-180
Applications of Zagier’s Formula (I)....Pages 181-202
Applications of Zagier’s Formula (II)....Pages 203-230
Back Matter....Pages 231-256
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