Bernoulli numbers and zeta functions:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Tokyo [u.a.]
Springer
2014
|
| Series: | Springer Monographs in Mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XI, 274 S. graph. Darst. |
| ISBN: | 9784431549185 |
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| adam_text |
Springer
Monographs in Mathematics
Tsuneo Arakawa
·
Tomoyoshi Ibukiyama
·
Masanobu Kaneko
Bernoulli Numbers and
Zeta
Functions
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;
Kummers
congruence between Bernoulli numbers and a related theory of p-adic measures; the
Euler-Madaurin
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 new.
Contents
Bernoulli Numbers
. 1
1.1
Definitions: Introduction from History
. 1
1.2
Sums of Consecutive Powers of Integers
and Theorem of Faulhaber
. 6
1.3
Forma) Power Series
. 13
1.4
The Generating Function of Bernoulli Numbers
. 20
Stirling Numbers and Bernoulli Numbers
. 25
2.1
Stirling Numbers
. 25
2.2
Formulas for the Bernoulli Numbers Involving
the Stirling Numbers
. 34
Theorem of Clausen and
von Staudt,
and Rummer's
Congruence
. 41
3.1
Theorem of Clausen and
von
Staudt
. 41
3.2
Rummer's Congruence
. 43
3.3
Short Biographies of Clausen,
von
Staudt and
Kummer. 46
Generalized Bernoulli Numbers
. 51
4.
1 Dirichlet Characters
. 51
4.2
Generalized Bernoulli Numbers
. 53
4.3
Bernoulli Polynomials
. 55
The Euler-Maclaurin Summation Formula and the
Riemann
Zeta
Function
. 65
5.1
Euler-Maclaurin Summation Formula
. 65
5.2
The Riemann
Zeta
Function
. 67
Quadratic Forms and Ideal Theory of Quadratic Fields
. 75
6.1
Quadratic Forms
. 75
6.2
Orders of Quadratic Fields
. 77
6.3
Class Number Formula of Quadratic Forms
. 87
IX
χ
Contents
7
Congruence Between Bernoulli Numbers
and Class Numbers of Imaginary Quadratic Fields
. 95
7.1
Congruence Between Bernoulli Numbers and Class Numbers
_ 95
7.2
"Hurwitz-integral" Series
. 97
7.3
Proof of Theorem
7.1 . 99
8
Character Sums and Bernoulli Numbers
. 103
8.1
Simplest Examples
. 104
8.2
Gaussian Sum
. 107
8.3
Exponential Sums and Generalized Bernoulli Numbers
. 110
8.4
Various Examples of Sums
. 118
8.5
Sporadic Examples: Using Functions
. 121
8.6
Sporadic Examples: Using the Symmetry
. 122
8.7
Sporadic Example: Symmetrize Asymmetry
. 127
8.8
Quadratic Polynomials and Character Sums
. 132
8.9
A Sum with Quadratic Conditions
. 133
9
Special Values and Complex Integral Representation
of ¿-Functions
. 139
9.1
The Hurwitz
Zeta
Function
.
J
39
9.2
Contour Integral
. 141
9.3
The Functional Equation of £(.v, a)
. 145
9.4
Special Values of L-Functions and the Functional Equations
. 148
10
Class Number Formula and an Easy
Zeta
Function
of the Space of Quadratic Forms
. 155
10.1
Ideal Class Groups of Quadratic Fields
. 155
10.2
Proof of the Class Number Formula of Imaginary
Quadratic Fields
. 162
10.3
Some
L
-Functions Associated with Quadratic Forms
. 173
П
/7-adic Measure and Rummer's Congruence
. 183
11.1
Measure on the Ring of /?-adic Integers and the Ring
of Formal Power Series
. 183
11.2
Bernoulli Measure
. 196
1
1.3
Rummer's Congruence Revisited
. 198
12
Hurwitz Numbers
. 203
12.1
Hurwitz Numbers
. 203
12.2
A Short Biography of Hurwitz
. 207
13
The Barnes Multiple
Zeta
Function
. 209
13.1
Special Values of Multiple
Zeta
Functions
and Bernoulli Polynomials
. 210
1
3.2
The Double
Zeta
Functions and Dirichlet Series
. 212
13.3
ξ(Μ.α)
and Continued Fractions
. 218
Contents xi
14
Poly-Bernoulli Numbers
. 223
14.1
Poly-Bernoulli Numbers
. 223
14.2
Theorem of Clausen and
von
Staudt Type
. 227
14.3
Poly-Bernoulli Numbers with Negative Upper Indices
. 233
Appendix (by Don Zagier): Curious and Exotic Identities for
Bernoulli Numbers
. 239
A.I The 'Other" Generating
Function(s)
for the Bernoulli
Numbers
. 240
A.
2
An Application: Periodicity of Modified Bernoulli Numbers
. 244
A.3
Miki
's
Identity
. 246
A.4 Products and Scalar Products of Bernoulli Polynomials
. 249
A.
5
Continued Fraction Expansions for Generating
Functions of Bernoulli Numbers
. 256
References
. 263
Index
. 269 |
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| author | Arakawa, Tsuneo 1949-2003 Ibukiyama, Tomoyoshi Kaneko, Masanobu |
| author_GND | (DE-588)113768745 (DE-588)1058624040 (DE-588)1058624059 |
| author_facet | Arakawa, Tsuneo 1949-2003 Ibukiyama, Tomoyoshi Kaneko, Masanobu |
| author_role | aut aut aut |
| author_sort | Arakawa, Tsuneo 1949-2003 |
| author_variant | t a ta t i ti m k mk |
| building | Verbundindex |
| bvnumber | BV042026826 |
| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)891349538 (DE-599)BVBBV042026826 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV042026826 |
| illustrated | Illustrated |
| indexdate | 2025-03-03T13:02:22Z |
| institution | BVB |
| isbn | 9784431549185 |
| language | English |
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| physical | XI, 274 S. graph. Darst. |
| publishDate | 2014 |
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| publisher | Springer |
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| series2 | Springer Monographs in Mathematics |
| spelling | Arakawa, Tsuneo 1949-2003 Verfasser (DE-588)113768745 aut Bernoulli numbers and zeta functions Tsuneo Arakawa ; Tomoyoshi Ibukiyama ; Masanobu Kaneko Tokyo [u.a.] Springer 2014 XI, 274 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer Monographs in Mathematics Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zetafunktion (DE-588)4190764-4 gnd rswk-swf Bernoullische Zahl (DE-588)4276648-5 gnd rswk-swf Bernoullische Zahl (DE-588)4276648-5 s Zetafunktion (DE-588)4190764-4 s Zahlentheorie (DE-588)4067277-3 s DE-604 Ibukiyama, Tomoyoshi Verfasser (DE-588)1058624040 aut Kaneko, Masanobu Verfasser (DE-588)1058624059 aut Erscheint auch als Online-Ausgabe 978-4-431-54919-2 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027468361&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027468361&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Arakawa, Tsuneo 1949-2003 Ibukiyama, Tomoyoshi Kaneko, Masanobu Bernoulli numbers and zeta functions Zahlentheorie (DE-588)4067277-3 gnd Zetafunktion (DE-588)4190764-4 gnd Bernoullische Zahl (DE-588)4276648-5 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4190764-4 (DE-588)4276648-5 |
| title | Bernoulli numbers and zeta functions |
| title_auth | Bernoulli numbers and zeta functions |
| title_exact_search | Bernoulli numbers and zeta functions |
| title_full | Bernoulli numbers and zeta functions Tsuneo Arakawa ; Tomoyoshi Ibukiyama ; Masanobu Kaneko |
| title_fullStr | Bernoulli numbers and zeta functions Tsuneo Arakawa ; Tomoyoshi Ibukiyama ; Masanobu Kaneko |
| title_full_unstemmed | Bernoulli numbers and zeta functions Tsuneo Arakawa ; Tomoyoshi Ibukiyama ; Masanobu Kaneko |
| title_short | Bernoulli numbers and zeta functions |
| title_sort | bernoulli numbers and zeta functions |
| topic | Zahlentheorie (DE-588)4067277-3 gnd Zetafunktion (DE-588)4190764-4 gnd Bernoullische Zahl (DE-588)4276648-5 gnd |
| topic_facet | Zahlentheorie Zetafunktion Bernoullische Zahl |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027468361&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027468361&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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