Experimental number theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford [u.a.]
Oxford University Press
2007
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Oxford graduate texts in mathematics
13 |
| Schlagworte: | |
| Online-Zugang: | Table of contents Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references (p. [205]-210) and index |
| Beschreibung: | xvi, 214 p. 24 cm |
| ISBN: | 9780199227303 9780198528227 |
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| 245 | 1 | 0 | |a Experimental number theory |c Fernando Rodriguez Villegas |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Oxford [u.a.] |b Oxford University Press |c 2007 | |
| 300 | |a xvi, 214 p. |c 24 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Oxford graduate texts in mathematics |v 13 | |
| 500 | |a Includes bibliographical references (p. [205]-210) and index | ||
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Number theory |x Data processing | |
| 650 | 4 | |a Number theory |x Data processing |v Problems, exercises, etc | |
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Datensatz im Suchindex
| _version_ | 1804137528995872768 |
|---|---|
| adam_text | Contents
1 Basic
examples
1
1
1
5
6
7
10
10
14
15
19
19
20
21
23
27
28
28
29
31
36
36
40
2.1
More variation with p
40
2.1.1
Loca! zeta
functions
40
2.1.2
Formulation of reciprocity
41
2.1.3
Globa!
zeta
functions
43
1.1
How
t
hings vary with
p
1.1.1
Quadratic Reciprocity Law
1.1.2
Sign functions
1.1.3
Checking modularity of sign functions
1.1.4
Examples
1.2
Recos
jnizing numbers
1.2.1
Rational numbers in
R
1.2.2
Rational numbers modulo
m
1.2.3
Algebraic numbers
1.3
Bernoulli polynomials
1.3.1
Definition and properties
1.3.2
Calculation
1.3.3
Related questions
1.3.4
Proofs
1.4
Sums
of squares
1.4.1
łc=1
1.4.2
k = 2
1.4.3
k>3
1.4.4
The numbers
г^(п)
1.4.5
Theta functions
1.5
Exercises
Reciprocitv
Contents
2.2
The cubic case
44
2.2.1
Two examples
45
2.2.2
A modular case
46
2.2.3
A non-modular case
48
2.3
The
Artin
map
50
2.3.1
A Galois example
50
2.3.2
A non-Galois example
53
2.4
Quantitative version
55
2.4.1
Application of a theorem of Tchebotarev
55
2.4.2
Computing densities numerically
56
2.5
Galois groups
59
2.5.1
Tchebotarev s theorem
63
2.5.2
Trink s example
64
2.5.3
An example related to Trink s
65
2.6
Exercises
69
3
Positive definite binary quadratic forms
71
3.1
3.2
Basic facts
71
3.1.1
Reduction
72
3.1.2
Comachia s algorithm
73
3.1.3
Class number
75
3.1.4
Composition
76
Examples of reciprocity for imaginary quadratic fields
77
3.2.1
Dihedral group of order
6
77
3.2.2
Theta functions again
79
3.2.3
Dihedral group of order
10
82
3.2.4
An example of
F. Voloch
84
3.2.5
Final comments
90
Fxprrkpç
91
3.3
4
Sequences
92
4.1
Trinomial numbers
92
4.1.1
Formula
92
4.1.2
Differential equation and linear recurrence 93
4.1.3
Algebraic equation
98
4.1.4
Hensel s lemma and Newton s method
99
4.1.5
Continued fractions
103
4.1.6
Asymptotics
107
Contents
4.1.7
More coefficients in the asymptotic expansion
109
4.1.8
Can we sum the asymptotic series?
111
4.2
Recognizing sequences
113
4.2.1
Values of a polynomial
113
4.2.2
Values of a rational function
114
4.2.3
Constant term recursion
115
4.2.4
A simple example
117
4.3
Exercises
119
5
Combinatorics
123
5.1
Description of the basic algorithm
123
5.2
Partitions
127
5.2.1
The number of partitions
128
5.2.2
Dual partition
130
5.3
Irreducible representations of 5n
131
5.3.1
Hook formula
132
5.3.2
The Murnaghan-Nakayama rule
133
5.3.3
Counting solutions to equations in Sn
138
5.3.4
Counting homomorphism and subgroups
139
5.4
Cyclotomic polynomials
143
5.4.1
Values of
φ
below a given bound
143
5.4.2
Computing cyclotomic polynomials
145
5.5
Exercises
148
6
p-adic numbers
151
6.1
Basic functions
151
6.1.1
Mahler s expansion
151
6.1.2
Hensel s lemma and Newton s method (again)
152
6.2
The p-adic gamma function
155
6.2.1
The multiplication formula
158
6.3
The logarithmic derivative of
Гр
159
6.3.1
Application to harmonic sums
161
6.3.2
A formula of J. Diamond
163
6.3.3
Power series expansion of
ψρ(χ)
164
6.3.4
Application to congruences
167
Contents
6.4
Analytic
continuation
168
6.4.1
An
example of Dwork
169
6.4.2
A generalization
170
6.4.3
Dwork s exponential
170
6.5
Gauss sums and the Gross-Koblitz formula
172
6.5.1
The case of Fp
172
6.5.2
An example
174
6.6
Exercises
175
7
Polynomials
177
7.1
Mahler s measure
177
7.1.1
Simple search
178
7.1.2
Refining the search
179
7.1.3
Counting roots on the unit circle
182
7.2
Applications of the Graeffe map
182
7.2.1
Detecting cyclotomic polynomials
182
7.2.2
Detecting cyclotomic factors
184
7.2.3
Wedge product polynomial
184
7.2.4
Interlacing roots of unity
185
7.3
Exercises
188
8
Remarks on selected exercises
189
References
205
Index
211
|
| adam_txt |
Contents
1 Basic
examples
1
1
1
5
6
7
10
10
14
15
19
19
20
21
23
27
28
28
29
31
36
36
40
2.1
More variation with p
40
2.1.1
Loca! zeta
functions
40
2.1.2
Formulation of reciprocity
41
2.1.3
Globa!
zeta
functions
43
1.1
How
t
hings vary with
p
1.1.1
Quadratic Reciprocity Law
1.1.2
Sign functions
1.1.3
Checking modularity of sign functions
1.1.4
Examples
1.2
Recos
jnizing numbers
1.2.1
Rational numbers in
R
1.2.2
Rational numbers modulo
m
1.2.3
Algebraic numbers
1.3
Bernoulli polynomials
1.3.1
Definition and properties
1.3.2
Calculation
1.3.3
Related questions
1.3.4
Proofs
1.4
Sums
of squares
1.4.1
łc=1
1.4.2
k = 2
1.4.3
k>3
1.4.4
The numbers
г^(п)
1.4.5
Theta functions
1.5
Exercises
Reciprocitv
Contents
2.2
The cubic case
44
2.2.1
Two examples
45
2.2.2
A modular case
46
2.2.3
A non-modular case
48
2.3
The
Artin
map
50
2.3.1
A Galois example
50
2.3.2
A non-Galois example
53
2.4
Quantitative version
55
2.4.1
Application of a theorem of Tchebotarev
55
2.4.2
Computing densities numerically
56
2.5
Galois groups
59
2.5.1
Tchebotarev's theorem
63
2.5.2
Trink's example
64
2.5.3
An example related to Trink's
65
2.6
Exercises
69
3
Positive definite binary quadratic forms
71
3.1
3.2
Basic facts
71
3.1.1
Reduction
72
3.1.2
Comachia's algorithm
73
3.1.3
Class number
75
3.1.4
Composition
76
Examples of reciprocity for imaginary quadratic fields
77
3.2.1
Dihedral group of order
6
77
3.2.2
Theta functions again
79
3.2.3
Dihedral group of order
10
82
3.2.4
An example of
F. Voloch
84
3.2.5
Final comments
90
Fxprrkpç
91
3.3
4
Sequences
92
4.1
Trinomial numbers
92
4.1.1
Formula
92
4.1.2
Differential equation and linear recurrence 93
4.1.3
Algebraic equation
98
4.1.4
Hensel's lemma and Newton's method
99
4.1.5
Continued fractions
103
4.1.6
Asymptotics
107
Contents
4.1.7
More coefficients in the asymptotic expansion
109
4.1.8
Can we sum the asymptotic series?
111
4.2
Recognizing sequences
113
4.2.1
Values of a polynomial
113
4.2.2
Values of a rational function
114
4.2.3
Constant term recursion
115
4.2.4
A simple example
117
4.3
Exercises
119
5
Combinatorics
123
5.1
Description of the basic algorithm
123
5.2
Partitions
127
5.2.1
The number of partitions
128
5.2.2
Dual partition
130
5.3
Irreducible representations of 5n
131
5.3.1
Hook formula
132
5.3.2
The Murnaghan-Nakayama rule
133
5.3.3
Counting solutions to equations in Sn
138
5.3.4
Counting homomorphism and subgroups
139
5.4
Cyclotomic polynomials
143
5.4.1
Values of
φ
below a given bound
143
5.4.2
Computing cyclotomic polynomials
145
5.5
Exercises
148
6
p-adic numbers
151
6.1
Basic functions
151
6.1.1
Mahler's expansion
151
6.1.2
Hensel's lemma and Newton's method (again)
152
6.2
The p-adic gamma function
155
6.2.1
The multiplication formula
158
6.3
The logarithmic derivative of
Гр
159
6.3.1
Application to harmonic sums
161
6.3.2
A formula of J. Diamond
163
6.3.3
Power series expansion of
ψρ(χ)
164
6.3.4
Application to congruences
167
Contents
6.4
Analytic
continuation
168
6.4.1
An
example of Dwork
169
6.4.2
A generalization
170
6.4.3
Dwork's exponential
170
6.5
Gauss sums and the Gross-Koblitz formula
172
6.5.1
The case of Fp
172
6.5.2
An example
174
6.6
Exercises
175
7
Polynomials
177
7.1
Mahler's measure
177
7.1.1
Simple search
178
7.1.2
Refining the search
179
7.1.3
Counting roots on the unit circle
182
7.2
Applications of the Graeffe map
182
7.2.1
Detecting cyclotomic polynomials
182
7.2.2
Detecting cyclotomic factors
184
7.2.3
Wedge product polynomial
184
7.2.4
Interlacing roots of unity
185
7.3
Exercises
188
8
Remarks on selected exercises
189
References
205
Index
211 |
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| id | DE-604.BV023234627 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T20:21:27Z |
| indexdate | 2024-07-09T21:13:44Z |
| institution | BVB |
| isbn | 9780199227303 9780198528227 |
| language | English |
| lccn | 2007299209 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016420263 |
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| physical | xvi, 214 p. 24 cm |
| publishDate | 2007 |
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| publisher | Oxford University Press |
| record_format | marc |
| series | Oxford graduate texts in mathematics |
| series2 | Oxford graduate texts in mathematics |
| spelling | Villegas, Fernando Rodriguez Verfasser aut Experimental number theory Fernando Rodriguez Villegas 1. publ. Oxford [u.a.] Oxford University Press 2007 xvi, 214 p. 24 cm txt rdacontent n rdamedia nc rdacarrier Oxford graduate texts in mathematics 13 Includes bibliographical references (p. [205]-210) and index Datenverarbeitung Number theory Data processing Number theory Data processing Problems, exercises, etc Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Oxford graduate texts in mathematics 13 (DE-604)BV011416591 13 http://www.loc.gov/catdir/toc/fy0801/2007299209.html Table of contents Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016420263&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Villegas, Fernando Rodriguez Experimental number theory Oxford graduate texts in mathematics Datenverarbeitung Number theory Data processing Number theory Data processing Problems, exercises, etc Zahlentheorie (DE-588)4067277-3 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4128130-5 |
| title | Experimental number theory |
| title_auth | Experimental number theory |
| title_exact_search | Experimental number theory |
| title_exact_search_txtP | Experimental number theory |
| title_full | Experimental number theory Fernando Rodriguez Villegas |
| title_fullStr | Experimental number theory Fernando Rodriguez Villegas |
| title_full_unstemmed | Experimental number theory Fernando Rodriguez Villegas |
| title_short | Experimental number theory |
| title_sort | experimental number theory |
| topic | Datenverarbeitung Number theory Data processing Number theory Data processing Problems, exercises, etc Zahlentheorie (DE-588)4067277-3 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Datenverarbeitung Number theory Data processing Number theory Data processing Problems, exercises, etc Zahlentheorie Numerisches Verfahren |
| url | http://www.loc.gov/catdir/toc/fy0801/2007299209.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016420263&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011416591 |
| work_keys_str_mv | AT villegasfernandorodriguez experimentalnumbertheory |