Analytic number theory: exploring the anatomy of integers
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Math. Soc.
2012
|
| Schriftenreihe: | Graduate studies in mathematics
134 |
| Schlagworte: |
Number theory
> Elementary number theory
> Multiplicative structure; Euclidean algorithm; greatest common divisors
Number theory
> Sequences and sets
> Fibonacci and Lucas numbers and polynomials and generalizations
Number theory
> Probabilistic theory: distribution modulo 1; metric theory of algorithms
> Arithmetic functions
|
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 414 S. |
| ISBN: | 9780821875773 |
Internformat
MARC
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| 100 | 1 | |a Koninck, Jean-Marie de |d 1948- |e Verfasser |0 (DE-588)141191112 |4 aut | |
| 245 | 1 | 0 | |a Analytic number theory |b exploring the anatomy of integers |c Jean-Marie De Koninck ; Florian Luca |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2012 | |
| 300 | |a XVIII, 414 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 134 | |
| 650 | 4 | |a Number theory | |
| 650 | 4 | |a Euclidean algorithm | |
| 650 | 4 | |a Integrals | |
| 650 | 7 | |a Number theory -- Elementary number theory -- Multiplicative structure; Euclidean algorithm; greatest common divisors |2 msc | |
| 650 | 7 | |a Number theory -- Elementary number theory -- Primes |2 msc | |
| 650 | 7 | |a Number theory -- Sequences and sets -- Density, gaps, topology |2 msc | |
| 650 | 7 | |a Number theory -- Sequences and sets -- Fibonacci and Lucas numbers and polynomials and generalizations |2 msc | |
| 650 | 7 | |a Number theory -- Probabilistic theory: distribution modulo 1; metric theory of algorithms -- Arithmetic functions |2 msc | |
| 650 | 7 | |a Number theory -- Multiplicative number theory -- Distribution of primes |2 msc | |
| 650 | 7 | |a Number theory -- Multiplicative number theory -- Primes in progressions |2 msc | |
| 650 | 7 | |a Number theory -- Multiplicative number theory -- Sieves |2 msc | |
| 650 | 7 | |a Number theory -- Multiplicative number theory -- Asymptotic results on arithmetic functions |2 msc | |
| 650 | 7 | |a Number theory -- Multiplicative number theory -- Distribution functions associated with additive and positive multiplicative functions |2 msc | |
| 650 | 0 | 7 | |a Analytische Zahlentheorie |0 (DE-588)4001870-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Analytische Zahlentheorie |0 (DE-588)4001870-2 |D s |
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| 700 | 1 | |a Luca, Florian |d 1969- |e Verfasser |0 (DE-588)1025704193 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-0-8218-8541-3 |
| 830 | 0 | |a Graduate studies in mathematics |v 134 |w (DE-604)BV009739289 |9 134 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024984372&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-024984372 | ||
Datensatz im Suchindex
| _version_ | 1804149090326413312 |
|---|---|
| adam_text | Contents
Preface
ix
Notation xiii
Frequently Used Functions
xvii
Chapter
1.
Preliminary Notions
1
§1.1.
Approximating a sum by an integral
1
§1.2.
The Euler-MacLaurin formula
2
§1.3.
The Abel summation formula
5
§1.4.
Stieltjes
integrals
7
§1.5.
Slowly oscillating functions
8
§1.6.
Combinatorial results
9
§1.7.
The Chinese Remainder Theorem
10
§1.8.
The density of a set of integers
11
§1.9.
The Stirling formula
11
§1.10.
Basic inequalities
13
Problems on Chapter
1 15
Chapter
2.
Prime Numbers and Their Properties
19
§2.1.
Prime numbers and their polynomial representations
19
§2.2.
There exist infinitely many primes
21
§2.3.
A first glimpse at the size of
π
(ж)
21
§2.4.
Fermat
numbers
22
§2.5.
A better lower bound for
π(χ)
24
iii
iv Contents
§2.6.
The Chebyshev estimates
24
§2.7.
The
Bertrand
Postulate
29
§2.8.
The distance between consecutive primes
31
§2.9.
Mersenne primes
32
§2.10.
Conjectures on the distribution of prime numbers
33
Problems on Chapter
2 36
Chapter
3.
The Riemann
Zeta
Function
39
§3.1.
The definition of the Riemann
Zeta
Function
39
§3.2.
Extending the
Zeta
Function to the half-plane
σ
> 0 40
§3.3.
The derivative of the Riemann
Zeta
Function
41
§3.4.
The zeros of the
Zeta
Function
43
§3.5.
Euler s estimate
ζ
(2) =
тг2/6
45
Problems on Chapter
3 48
Chapter
4.
Setting the Stage for the Proof of the Prime Number
Theorem
51
§4.1.
Key functions related to the Prime Number Theorem
51
§4.2.
A closer analysis of the functions
θ(χ)
and
ψ(χ)
52
§4.3.
Useful estimates
53
§4.4.
The Mertens estimate
55
§4.5.
The
Möbius
function
56
§4.6.
The divisor function
58
Problems on Chapter
4 60
Chapter
5.
The Proof of the Prime Number Theorem
63
§5.1.
A theorem of D. J. Newman
63
§5.2.
An application of Newman s theorem
65
§5.3.
The proof of the Prime Number Theorem
66
§5.4.
A review of the proof of the Prime Number Theorem
69
§5.5.
The Riemann Hypothesis and the Prime Number Theorem
70
§5.6.
Useful estimates involving primes
71
§5.7.
Elementary proofs of the Prime Number Theorem
72
Problems on Chapter
5 72
Chapter
6.
The Global Behavior of Arithmetic Functions
75
§6.1.
Dirichlet series and arithmetic functions
75
§6.2.
The uniqueness of representation of a Dirichlet series
77
Contents
§6.3.
Multiplicative
functions
79
§6.4.
Generating functions and Dirichlet products
81
§6.5.
Wintner s theorem
82
§6.6.
Additive functions
85
§6.7.
The average orders of
ω(η)
and
Ω(η)
86
§6.8.
The average order of an additive function
87
§6.9.
The
Erdős-
Wintner theorem
88
Problems on Chapter
6 89
Chapter
7.
The Local Behavior of Arithmetic Functions
93
§7.1.
The normal order of an arithmetic function
93
§7.2.
The
Turán-Kubilius
inequality
94
§7.3.
Maximal order of the divisor function
99
§7.4.
An upper bound for d(n)
101
§7.5.
Asymptotic densities
103
§7.6.
Perfect numbers
106
§7.7.
Sierpiński, Riesel,
and Romanov
106
§7.8.
Some open problems of an elementary nature
108
Problems on Chapter
7 109
Chapter
8.
The Fascinating
Euler
Function
115
§8.1.
The
Euler
function
115
§8.2.
Elementary properties of the
Euler
function
117
§8.3.
The average order of the
Euler
function
118
§8.4.
When is
φ(η)σ(η)
a square?
119
§8.5.
The distribution of the values of
φ(η)/η
121
§8.6.
The local behavior of the
Euler
function
122
Problems on Chapter
8 124
Chapter
9.
Smooth Numbers
127
§9.1.
Notation
127
§9.2.
The smallest prime factor of an integer
127
§9.3.
The largest prime factor of an integer
131
§9.4.
The Rankin method
137
§9.5.
An application to
pseudoprimes
141
§9.6.
The geometric method
145
§9.7.
The best known estimates on
Ф(ж, у)
146
vi
Contents
§9.8.
The
Dickman
function
147
§9.9.
Consecutive smooth numbers
149
Problems on Chapter
9 150
Chapter
10.
The Hardy-Ramanujan and Landau Theorems
157
§10.1.
The Hardy-Ramanujan inequality
157
§10.2.
Landau s theorem
159
Problems on Chapter
10 164
Chapter
11.
The
abc
Conjecture and Some of Its Applications
167
§11.1.
The
abc
conjecture
167
§11.2.
The relevance of the condition
ε
> 0 168
§11.3.
The Generalized
Fermat
Equation
171
§11.4.
Consecutive powerful numbers
172
§11.5.
Sums of ^-powerful numbers
172
§11.6.
The
Erdős-Woods
conjecture
173
§11.7.
A problem of Gandhi
174
§11.8.
The k-abc conjecture
175
Problems on Chapter
11 176
Chapter
12.
Sieve Methods
179
§12.1.
The sieve of Eratosthenes
179
§12.2.
The
Brun
sieve
180
§12.3.
Twin primes
184
§12.4.
The
Brun
combinatorial sieve
187
§12.5.
A Chebyshev type estimate
187
§12.6.
The Brun-Titchmarsh theorem
188
§12.7.
Twin primes revisited
190
§12.8.
Smooth shifted primes
191
§12.9.
The
Goldbach
conjecture
192
§12.10.
The Schnirelman theorem
194
§12.11.
The Selberg sieve
198
§12.12.
The Brun-Titchmarsh theorem from the Selberg sieve
201
§12.13.
The Large sieve
202
§12.14.
Quasi-squares
203
§12.15.
The smallest quadratic
nonresidue
modulo
ρ
204
Problems on Chapter
12 206
Contents
vii
Chapter
13.
Prime Numbers in Arithmetic Progression
217
§13.1.
Quadratic residues
217
§13.2.
The proof of the Quadratic Reciprocity Law
220
§13.3.
Primes in arithmetic progressions with small moduli
222
§13.4.
The Primitive Divisor theorem
224
§13.5.
Comments on the Primitive Divisor theorem
227
Problems on Chapter
13 228
Chapter
14.
Characters and the Dirichlet Theorem
233
§14.1.
Primitive roots
233
§14.2.
Characters
235
§14.3.
Theorems about characters
236
§14.4.
L-series
240
§14.5.
L(l,x) is finite if
χ
is a non-principal character
242
§14.6.
The nonvanishing of L(l,x): first step
243
§14.7.
The completion of the proof of the Dirichlet theorem
244
Problems on Chapter
14 247
Chapter
15.
Selected Applications of Primes in Arithmetic
Progression
251
§15.1.
Known results about primes in arithmetical progressions
251
§15.2.
Some Diophantine applications
254
§15.3.
Primes
ρ
with
ρ
— 1
squarefree
257
§15.4.
More applications of primes in arithmetic progressions
259
§15.5.
Probabilistic applications
261
Problems on Chapter
15 263
Chapter
16.
The Index of Composition of an Integer
267
§16.1.
Introduction
267
§16.2.
Elementary results
268
§16.3.
Mean values of A and I/A
270
§16.4.
Local behavior of A(n)
273
§16.5.
Distribution function of
λ(η)
275
§16.6.
Probabilistic results
276
Problems on Chapter
16 279
Appendix: Basic Complex Analysis Theory
281
§17.1.
Basic definitions
281
viii Contents
§17.2.
Infinite
products
283
§17.3.
The derivative of a function of a complex variable
284
§17.4.
The integral of a function along a path
285
§17.5.
The Cauchy theorem
287
§17.6.
The Cauchy integral formula
289
Solutions to Even-Numbered Problems
291
Solutions to problems from Chapter
1 291
Solutions to problems from Chapter
2 295
Solutions to problems from Chapter
3 303
Solutions to problems from Chapter
4 309
Solutions to problems from Chapter
5 312
Solutions to problems from Chapter
б
318
Solutions to problems from Chapter
7 321
Solutions to problems from Chapter
8 334
Solutions to problems from Chapter
9 338
Solutions to problems from Chapter
10 351
Solutions to problems from Chapter
11 353
Solutions to problems from Chapter
12 356
Solutions to problems from Chapter
13 377
Solutions to problems from Chapter
14 384
Solutions to problems from Chapter
15 392
Solutions to problems from Chapter
16 401
Bibliography
405
Index
413
|
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| author | Koninck, Jean-Marie de 1948- Luca, Florian 1969- |
| author_GND | (DE-588)141191112 (DE-588)1025704193 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.7/4 |
| dewey-search | 512.7/4 |
| dewey-sort | 3512.7 14 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV040127117 |
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| indexdate | 2024-07-10T00:17:30Z |
| institution | BVB |
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| language | English |
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| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Koninck, Jean-Marie de 1948- Verfasser (DE-588)141191112 aut Analytic number theory exploring the anatomy of integers Jean-Marie De Koninck ; Florian Luca Providence, RI American Math. Soc. 2012 XVIII, 414 S. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 134 Number theory Euclidean algorithm Integrals Number theory -- Elementary number theory -- Multiplicative structure; Euclidean algorithm; greatest common divisors msc Number theory -- Elementary number theory -- Primes msc Number theory -- Sequences and sets -- Density, gaps, topology msc Number theory -- Sequences and sets -- Fibonacci and Lucas numbers and polynomials and generalizations msc Number theory -- Probabilistic theory: distribution modulo 1; metric theory of algorithms -- Arithmetic functions msc Number theory -- Multiplicative number theory -- Distribution of primes msc Number theory -- Multiplicative number theory -- Primes in progressions msc Number theory -- Multiplicative number theory -- Sieves msc Number theory -- Multiplicative number theory -- Asymptotic results on arithmetic functions msc Number theory -- Multiplicative number theory -- Distribution functions associated with additive and positive multiplicative functions msc Analytische Zahlentheorie (DE-588)4001870-2 gnd rswk-swf Analytische Zahlentheorie (DE-588)4001870-2 s DE-604 Luca, Florian 1969- Verfasser (DE-588)1025704193 aut Erscheint auch als Online-Ausgabe 978-0-8218-8541-3 Graduate studies in mathematics 134 (DE-604)BV009739289 134 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024984372&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Koninck, Jean-Marie de 1948- Luca, Florian 1969- Analytic number theory exploring the anatomy of integers Graduate studies in mathematics Number theory Euclidean algorithm Integrals Number theory -- Elementary number theory -- Multiplicative structure; Euclidean algorithm; greatest common divisors msc Number theory -- Elementary number theory -- Primes msc Number theory -- Sequences and sets -- Density, gaps, topology msc Number theory -- Sequences and sets -- Fibonacci and Lucas numbers and polynomials and generalizations msc Number theory -- Probabilistic theory: distribution modulo 1; metric theory of algorithms -- Arithmetic functions msc Number theory -- Multiplicative number theory -- Distribution of primes msc Number theory -- Multiplicative number theory -- Primes in progressions msc Number theory -- Multiplicative number theory -- Sieves msc Number theory -- Multiplicative number theory -- Asymptotic results on arithmetic functions msc Number theory -- Multiplicative number theory -- Distribution functions associated with additive and positive multiplicative functions msc Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| subject_GND | (DE-588)4001870-2 |
| title | Analytic number theory exploring the anatomy of integers |
| title_auth | Analytic number theory exploring the anatomy of integers |
| title_exact_search | Analytic number theory exploring the anatomy of integers |
| title_full | Analytic number theory exploring the anatomy of integers Jean-Marie De Koninck ; Florian Luca |
| title_fullStr | Analytic number theory exploring the anatomy of integers Jean-Marie De Koninck ; Florian Luca |
| title_full_unstemmed | Analytic number theory exploring the anatomy of integers Jean-Marie De Koninck ; Florian Luca |
| title_short | Analytic number theory |
| title_sort | analytic number theory exploring the anatomy of integers |
| title_sub | exploring the anatomy of integers |
| topic | Number theory Euclidean algorithm Integrals Number theory -- Elementary number theory -- Multiplicative structure; Euclidean algorithm; greatest common divisors msc Number theory -- Elementary number theory -- Primes msc Number theory -- Sequences and sets -- Density, gaps, topology msc Number theory -- Sequences and sets -- Fibonacci and Lucas numbers and polynomials and generalizations msc Number theory -- Probabilistic theory: distribution modulo 1; metric theory of algorithms -- Arithmetic functions msc Number theory -- Multiplicative number theory -- Distribution of primes msc Number theory -- Multiplicative number theory -- Primes in progressions msc Number theory -- Multiplicative number theory -- Sieves msc Number theory -- Multiplicative number theory -- Asymptotic results on arithmetic functions msc Number theory -- Multiplicative number theory -- Distribution functions associated with additive and positive multiplicative functions msc Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| topic_facet | Number theory Euclidean algorithm Integrals Number theory -- Elementary number theory -- Multiplicative structure; Euclidean algorithm; greatest common divisors Number theory -- Elementary number theory -- Primes Number theory -- Sequences and sets -- Density, gaps, topology Number theory -- Sequences and sets -- Fibonacci and Lucas numbers and polynomials and generalizations Number theory -- Probabilistic theory: distribution modulo 1; metric theory of algorithms -- Arithmetic functions Number theory -- Multiplicative number theory -- Distribution of primes Number theory -- Multiplicative number theory -- Primes in progressions Number theory -- Multiplicative number theory -- Sieves Number theory -- Multiplicative number theory -- Asymptotic results on arithmetic functions Number theory -- Multiplicative number theory -- Distribution functions associated with additive and positive multiplicative functions Analytische Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024984372&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT koninckjeanmariede analyticnumbertheoryexploringtheanatomyofintegers AT lucaflorian analyticnumbertheoryexploringtheanatomyofintegers |