Congruences for L-Functions

J. Urbanowicz

ISBN 10: 9048154901 / ISBN 13: 9789048154906

Published by Springer, Springer, 2010

About this Item

Druck auf Anfrage Neuware - Printed after ordering - 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.

Seller Inventory # 9789048154906

Bibliographic Details

• Title: Congruences for L-Functions
• Publisher: Springer, Springer
• Publication Date: 2010
• Language: English
• ISBN 10: 9048154901
• ISBN 13: 9789048154906
• Binding: Taschenbuch
• Condition: Neu
• Dimensions: 235W x 155H x 15D millimeters 235 millimeters width by 155 millimeters height by 15 millimeters depth
• Weight: 417 grams

About this title

Synopsis

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

From the Back Cover

This book contains substantially extended and revised versions of the best papers from the 12th International Conference on Enterprise Information Systems (ICEIS 2010), held in Funchal, Madeira, Portugal, June 8-12, 2010.

Two invited papers are presented together with 39 contributions, which were carefully reviewed and selected from 62 full papers presented at the conference (out of 448 submissions). They reflect state-of-the-art research work that is often driven by real-world applications, thus successfully relating the academic with the industrial community. The topics covered are: databases and information systems integration, artificial intelligence and decision support systems, information systems analysis and specification, software agents and internet computing, and human-computer interaction.