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Bernoulli Numbers and Zeta Functions / by Tsuneo Arakawa, Tomoyoshi Ibukiyama, Masanobu Kaneko
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Springer Monographs in Mathematics | Springer eBook Collection | SpringerLink BücherPublisher: Tokyo [u.a.] : Springer, 2014Description: Online-Ressource (XI, 274 p. 5 illus., 1 illus. in color, online resource)ISBN:- 9784431549192
- 512.7
- QA241-247.5
Contents:
Summary: Two major subjects are treated in this book. The main one is the theory of Bernoulli numbers and the other is the theory of zeta functions. Historically, Bernoulli numbers were introduced to give formulas for the sums of powers of consecutive integers. The real reason that they are indispensable for number theory, however, lies in the fact that special values of the Riemann zeta function can be written by using Bernoulli numbers. This leads to more advanced topics, a number of which are treated in this book: Historical remarks on Bernoulli numbers and the formula for the sum of powers of consecutive integers; a formula for Bernoulli numbers by Stirling numbers; the Clausen-von Staudt theorem on the denominators of Bernoulli numbers; Kummer's congruence between Bernoulli numbers and a related theory of p-adic measures; the Euler-Maclaurin summation formula; the functional equation of the Riemann zeta function and the Dirichlet L functions, and their special values at suitable integers; various formulas of exponential sums expressed by generalized Bernoulli numbers; the relation between ideal classes of orders of quadratic fields and equivalence classes of binary quadratic forms; class number formula for positive definite binary quadratic forms; congruences between some class numbers and Bernoulli numbers; simple zeta functions of prehomogeneous vector spaces; Hurwitz numbers; Barnes multiple zeta functions and their special values; the functional equation of the double zeta functions; and poly-Bernoulli numbers. An appendix by Don Zagier on curious and exotic identities for Bernoulli numbers is also supplied. This book will be enjoyable both for amateurs and for professional researchers. Because the logical relations between the chapters are loosely connected, readers can start with any chapter depending on their interests. The expositions of the topics are not always typical, and some parts are completely newPPN: PPN: 1658921690Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Preface; Contents; 1 Bernoulli Numbers ; 1.1 Definitions: Introduction from History; 1.2 Sums of Consecutive Powers of Integers and Theorem of Faulhaber; 1.3 Formal Power Series; 1.4 The Generating Function of Bernoulli Numbers ; 2 Stirling Numbers and Bernoulli Numbers; 2.1 Stirling Numbers; 2.2 Formulas for the Bernoulli Numbers Involving the Stirling Numbers; 3 Theorem of Clausen and von Staudt, and Kummer's Congruence ; 3.1 Theorem of Clausen and von Staudt; 3.2 Kummer's Congruence; 3.3 Short Biographies of Clausen, von Staudt and Kummer; 4 Generalized Bernoulli Numbers
4.1 Dirichlet Characters4.2 Generalized Bernoulli Numbers; 4.3 Bernoulli Polynomials; 5 The Euler-Maclaurin Summation Formula and the Riemann Zeta Function; 5.1 Euler-Maclaurin Summation Formula; 5.2 The Riemann Zeta Function; 6 Quadratic Forms and Ideal Theory of Quadratic Fields; 6.1 Quadratic Forms; 6.2 Orders of Quadratic Fields; 6.3 Class Number Formula of Quadratic Forms; 7 Congruence Between Bernoulli Numbers and Class Numbers of Imaginary Quadratic Fields ; 7.1 Congruence Between Bernoulli Numbers and Class Numbers; 7.2 ``Hurwitz-integral'' Series; 7.3 Proof of Theorem 7.1
8 Character Sums and Bernoulli Numbers8.1 Simplest Examples; 8.2 Gaussian Sum; 8.3 Exponential Sums and Generalized Bernoulli Numbers; 8.4 Various Examples of Sums; 8.5 Sporadic Examples: Using Functions; 8.6 Sporadic Examples: Using the Symmetry; 8.7 Sporadic Example: Symmetrize Asymmetry; 8.8 Quadratic Polynomials and Character Sums; 8.9 A Sum with Quadratic Conditions; 9 Special Values and Complex Integral Representationof L-Functions; 9.1 The Hurwitz Zeta Function; 9.2 Contour Integral; 9.3 The Functional Equation of ζ(s,a); 9.4 Special Values of L-Functions and the Functional Equations
10 Class Number Formula and an Easy Zeta Function of the Space of Quadratic Forms10.1 Ideal Class Groups of Quadratic Fields; 10.2 Proof of the Class Number Formula of Imaginary Quadratic Fields; 10.3 Some L-Functions Associated with Quadratic Forms; 11 p-adic Measure and Kummer's Congruence; 11.1 Measure on the Ring of p-adic Integers and the Ring of Formal Power Series; 11.2 Bernoulli Measure; 11.3 Kummer's Congruence Revisited; 12 Hurwitz Numbers; 12.1 Hurwitz Numbers; 12.2 A Short Biography of Hurwitz; 13 The Barnes Multiple Zeta Function
13.1 Special Values of Multiple Zeta Functions and Bernoulli Polynomials13.2 The Double Zeta Functions and Dirichlet Series; 13.3 ξ(s,α) and Continued Fractions; 14 Poly-Bernoulli Numbers; 14.1 Poly-Bernoulli Numbers; 14.2 Theorem of Clausen and von Staudt Type; 14.3 Poly-Bernoulli Numbers with Negative Upper Indices; Appendix (by Don Zagier): Curious and Exotic Identities for Bernoulli Numbers; A.1 The ``Other'' Generating Function(s) for the BernoulliNumbers; A.2 An Application: Periodicity of Modified Bernoulli Numbers; A.3 Miki's Identity
A.4 Products and Scalar Products of Bernoulli Polynomials
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