Number theory in the spirit of Ramanujan:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2006]
|
| Schriftenreihe: | Student mathematical library
volume 34 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xix, 187 Seiten |
| ISBN: | 0821841785 9780821841785 |
Internformat
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| 245 | 1 | 0 | |a Number theory in the spirit of Ramanujan |c Bruce C. Berndt |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2006] | |
| 264 | 4 | |c © 2006 | |
| 300 | |a xix, 187 Seiten | ||
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| 490 | 1 | |a Student mathematical library |v volume 34 | |
| 600 | 1 | 4 | |a Ramanujan Aiyangar, Srinivasa <1887-1920> |
| 600 | 1 | 4 | |a Ramanujan Aiyangar, Srinivasa <1887-1920> |
| 650 | 4 | |a Fonctions thêta | |
| 650 | 7 | |a Getaltheorie |2 gtt | |
| 650 | 4 | |a Nombres, Théorie des | |
| 650 | 4 | |a Séries Q | |
| 650 | 7 | |a Theta-functies |2 gtt | |
| 650 | 4 | |a Functions, Theta | |
| 650 | 4 | |a Number theory | |
| 650 | 4 | |a q-series | |
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| 650 | 0 | 7 | |a Zahlentheorie |0 (DE-588)4067277-3 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-014943034 | ||
Datensatz im Suchindex
| _version_ | 1804135575532339200 |
|---|---|
| adam_text | Contents
Preface
ix
Chapter
1.
Introduction
1
§1.1.
Notation and Arithmetical Functions
1
§1.2.
What are g-Series and Theta Functions?
6
§1.3.
Fundamental Theorems about q-Series and Theta
Functions
7
§1.4.
Notes
22
Chapter
2.
Congruences for p(n) and
т(п)
27
§2.1.
Historical Background
27
§2.2.
Elementary Congruences for r(n)
28
§2.3.
Ramanujan s Congruence p(5n
+ 4)
ξ
0
(mod
5) 31
§2.4.
Ramanujan s Congruence p(7n
+ 5)
ξ
0
(mod
7) 39
§2.5.
The Parity of p(n)
43
§2.6.
Notes
49
Chapter
3.
Sums of Squares and Sums of Triangular Numbers
55
§3.1.
Lambert Series
55
§3.2.
Sums of Two Squares
56
§3.3.
Sums of Four Squares
59
vi
B. C. BERNDT
§3.4.
Sums of Six Squares
63
§3.5.
Sums of Eight Squares 67
§3.6.
Sums of Triangular Numbers
71
§3.7.
Representations of Integers by x2
+
2y2, x2
+
Зу2,
and
x2
+
xy
+
y2 72
§3.8.
Notes 79
Chapter
4. Eisenstein
Series
85
§4.1.
Bernoulli Numbers and
Eisenstein
Series
85
§4.2.
Trigonometric Series 87
§4.3.
A Class of Series from Ramanujan s Lost Notebook
Expressible in Terms of P, Q, and
R
97
§4.4.
Proofs of the Congruences p(5n
+ 4) = 0
(mod
5)
and
p(7n + 5) = 0(mod7) 102
§4.5.
Notes
105
Chapter
5.
The Connection Between Hypergeometric Functions
and Theta Functions 109
§5.1.
Definitions of Hypergeometric Series and Elliptic
Integrals
109
§5.2.
The Main Theorem H4
§5.3.
Principles of Duplication and Dimidiation
120
§5.4.
A Catalogue of Formulas for Theta Functions and
Eisenstein
Series
122
§5.5.
Notes
128
Chapter
6.
Applications of the Primary Theorem of Chapter
5 133
§6.1.
Introduction
133
§6.2.
Sums of Squares and Triangular Numbers
134
§6.3.
Modular Equations
140
§6.4.
Notes
150
Chapter
7.
The Rogers-Ramanujan Continued Fraction
153
§7.1.
Definition and Historical Background
153
SPIRIT OF RAMANUJAN
vii
§7.2.
The Convergence, Divergence, and Values of R(q)
155
§7.3.
The Rogers-Ramanujan Functions
158
§7.4.
Identities for R(q)
161
§7.5.
Modular Equations for R(q)
166
§7.6.
Notes
167
Bibliography
171
Index
185
|
| adam_txt |
Contents
Preface
ix
Chapter
1.
Introduction
1
§1.1.
Notation and Arithmetical Functions
1
§1.2.
What are g-Series and Theta Functions?
6
§1.3.
Fundamental Theorems about q-Series and Theta
Functions
7
§1.4.
Notes
22
Chapter
2.
Congruences for p(n) and
т(п)
27
§2.1.
Historical Background
27
§2.2.
Elementary Congruences for r(n)
28
§2.3.
Ramanujan's Congruence p(5n
+ 4)
ξ
0
(mod
5) 31
§2.4.
Ramanujan's Congruence p(7n
+ 5)
ξ
0
(mod
7) 39
§2.5.
The Parity of p(n)
43
§2.6.
Notes
49
Chapter
3.
Sums of Squares and Sums of Triangular Numbers
55
§3.1.
Lambert Series
55
§3.2.
Sums of Two Squares
56
§3.3.
Sums of Four Squares
59
vi
B. C. BERNDT
§3.4.
Sums of Six Squares
63
§3.5.
Sums of Eight Squares 67
§3.6.
Sums of Triangular Numbers
71
§3.7.
Representations of Integers by x2
+
2y2, x2
+
Зу2,
and
x2
+
xy
+
y2 72
§3.8.
Notes 79
Chapter
4. Eisenstein
Series
85
§4.1.
Bernoulli Numbers and
Eisenstein
Series
85
§4.2.
Trigonometric Series 87
§4.3.
A Class of Series from Ramanujan's Lost Notebook
Expressible in Terms of P, Q, and
R
97
§4.4.
Proofs of the Congruences p(5n
+ 4) = 0
(mod
5)
and
p(7n + 5) = 0(mod7) 102
§4.5.
Notes
105
Chapter
5.
The Connection Between Hypergeometric Functions
and Theta Functions 109
§5.1.
Definitions of Hypergeometric Series and Elliptic
Integrals
109
§5.2.
The Main Theorem H4
§5.3.
Principles of Duplication and Dimidiation
120
§5.4.
A Catalogue of Formulas for Theta Functions and
Eisenstein
Series
122
§5.5.
Notes
128
Chapter
6.
Applications of the Primary Theorem of Chapter
5 133
§6.1.
Introduction
133
§6.2.
Sums of Squares and Triangular Numbers
134
§6.3.
Modular Equations
140
§6.4.
Notes
150
Chapter
7.
The Rogers-Ramanujan Continued Fraction
153
§7.1.
Definition and Historical Background
153
SPIRIT OF RAMANUJAN
vii
§7.2.
The Convergence, Divergence, and Values of R(q)
155
§7.3.
The Rogers-Ramanujan Functions
158
§7.4.
Identities for R(q)
161
§7.5.
Modular Equations for R(q)
166
§7.6.
Notes
167
Bibliography
171
Index
185 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T15:25:53Z |
| indexdate | 2024-07-09T20:42:41Z |
| institution | BVB |
| isbn | 0821841785 9780821841785 |
| language | English |
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| physical | xix, 187 Seiten |
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| spelling | Berndt, Bruce C. 1939- Verfasser (DE-588)133753441 aut Number theory in the spirit of Ramanujan Bruce C. Berndt Providence, Rhode Island American Mathematical Society [2006] © 2006 xix, 187 Seiten txt rdacontent n rdamedia nc rdacarrier Student mathematical library volume 34 Ramanujan Aiyangar, Srinivasa <1887-1920> Fonctions thêta Getaltheorie gtt Nombres, Théorie des Séries Q Theta-functies gtt Functions, Theta Number theory q-series Thetafunktion (DE-588)4185175-4 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s Thetafunktion (DE-588)4185175-4 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-2145-8 Student mathematical library volume 34 (DE-604)BV013184751 34 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014943034&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Berndt, Bruce C. 1939- Number theory in the spirit of Ramanujan Student mathematical library Ramanujan Aiyangar, Srinivasa <1887-1920> Fonctions thêta Getaltheorie gtt Nombres, Théorie des Séries Q Theta-functies gtt Functions, Theta Number theory q-series Thetafunktion (DE-588)4185175-4 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4185175-4 (DE-588)4067277-3 |
| title | Number theory in the spirit of Ramanujan |
| title_auth | Number theory in the spirit of Ramanujan |
| title_exact_search | Number theory in the spirit of Ramanujan |
| title_exact_search_txtP | Number theory in the spirit of Ramanujan |
| title_full | Number theory in the spirit of Ramanujan Bruce C. Berndt |
| title_fullStr | Number theory in the spirit of Ramanujan Bruce C. Berndt |
| title_full_unstemmed | Number theory in the spirit of Ramanujan Bruce C. Berndt |
| title_short | Number theory in the spirit of Ramanujan |
| title_sort | number theory in the spirit of ramanujan |
| topic | Ramanujan Aiyangar, Srinivasa <1887-1920> Fonctions thêta Getaltheorie gtt Nombres, Théorie des Séries Q Theta-functies gtt Functions, Theta Number theory q-series Thetafunktion (DE-588)4185175-4 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Ramanujan Aiyangar, Srinivasa <1887-1920> Fonctions thêta Getaltheorie Nombres, Théorie des Séries Q Theta-functies Functions, Theta Number theory q-series Thetafunktion Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014943034&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013184751 |
| work_keys_str_mv | AT berndtbrucec numbertheoryinthespiritoframanujan |