Analytic number theory: an introductory course
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2004
|
| Edition: | 1. publ. |
| Series: | Monographs in number theory
1 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XV, 360 S. |
| ISBN: | 9812389385 9812560807 |
Staff View
MARC
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| 245 | 1 | 0 | |a Analytic number theory |b an introductory course |c Paul T. Bateman ; Harold G. Diamond |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2004 | |
| 300 | |a XV, 360 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Monographs in number theory |v 1 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 4 | |a Analytische Zahlentheorie | |
| 650 | 4 | |a Nombres, Théorie des | |
| 650 | 4 | |a Number theory | |
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| 700 | 1 | |a Diamond, Harold G. |d 1940- |e Verfasser |0 (DE-588)142256420 |4 aut | |
| 830 | 0 | |a Monographs in number theory |v 1 |w (DE-604)BV035341434 |9 1 | |
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Record in the Search Index
| _version_ | 1804138693395480576 |
|---|---|
| adam_text | Contents
Foreword
vii
Preface
ix
Chapter
1
Introduction
1
1.1
Three problems
........................... 1
1.2
Asymmetric distribution of quadratic residues
.......... 1
1.3
The prime number theorem
.................... 2
1.4
Density of squarefree integers
................... 3
1.5
The Riemann
zeta
function
.................... 8
1.6
Notes
................................ 11
Chapter
2
Calculus of Arithmetic Functions
13
2.1
Arithmetic functions and convolution
.............. 13
2.2
Inverses
............................... 17
2.3
Convergence
............................. 19
2.4
Exponential mapping
....................... 25
2.4.1
The
1
function as an exponential
............. 28
2.4.2
Powers and roots
...................... 29
2.5
Multiplicative functions
...................... 31
2.6
Notes
................................ 38
Chapter
3
Summatory Functions
39
3.1
Generalities
............................. 39
3.2
Estimate of Q(x)
-
бх/п2
..................... 42
3.3
Riemann-Stieltjes integrals
.................... 44
3.4
Riemann-Stieltjes integrators
................... 50
Xl
xü Contents
3.4.1
Convolution of integrators
................. 52
3.4.2
Generalization of results on arithmetic functions
.... 59
3.5
Stability
...............................
61
3.6
Dirichlet s hyperbola method
................... 66
3.7
Notes
................................
69
Chapter
4
The Distribution of Prime Numbers
71
4.1
General remarks
.......................... 71
4.2
The Chebyshev
φ
function
..................... 74
4.3
Mertens estimates
......................... 78
4.4
Convergent sums over primes
................... 81
4.5
A lower estimate for Euler s
ψ
function
............. 83
4.6
Notes
................................ 85
Chapter
5
An Elementary Proof of the P.N.T.
87
5.1
Selberg s formula
.......................... 87
5.1.1
Features of Selberg s formula
............... 90
5.2
Transformation of Selberg s formula
............... 91
5.2.1
Calculus for
R
....................... 92
5.3
Deduction of the P.N.T
....................... 96
5.4
Propositions equivalent to the P.N.T
.............. 98
5.5
Some consequences of the P.N.T
.................. 105
5.6
Notes
................................ 107
Chapter
6
Dirichlet Series and Mellin Transforms
109
6.1
The use of transforms
....................... 109
6.2
Euler
products
........................... 112
6.3
Convergence
............................. 116
6.3.1
Abscissa of convergence
.................. 118
6.3.2
Abscissa of absolute convergence
............. 120
6.4
Uniform convergence
........................ 120
6.5
Analyticity
............................. 125
6.5.1
Analytic continuation
................... 127
6.5.2
Continuation of
zeta
.................... 128
6.5.3
Example of analyticity on
σ
=
ac
............. 129
6.6
Uniqueness
............................. 129
6.6.1
Identifying an arithmetic function
............ 132
6.7
Operational calculus
........................ 133
Contents xiii
6.8
Landau s oscillation theorem.
................... 137
6.9
Notes
................................ 140
Chapter
7
Inversion Formulas
141
7.1
The use of inversion formulas
................... 141
7.2
The Wiener-Ikehara theorem
................... 143
7.2.1
Example. Counting product representations
....... 149
7.2.2
An O-estimate
....................... 151
7.3
A Wiener-Ikehara proof of the P.N.T
............... 151
7.4
A generalization of the Wiener-Ikehara theorem
......... 154
7.5
The Perron formula
........................ 162
7.6
Proof of the Perron formula
.................... 164
7.7
Contour deformation in the Perron formula
........... 168
7.7.1
The Fourier series of the sawtooth function
....... 169
7.7.2
Bounded and uniform convergence
............ 172
7.8
A smoothed Perron formula
................... 173
7.9
Example. Estimation of
£
Г(12
*
Із) ..............
176
7.10
Notes
................................ 180
Chapter
8
The Riemann
Zeta
Function
183
8.1
The functional equation
...................... 183
8.1.1
Justification of the interchange of ^ and
ƒ...... 185
8.1.2
Symmetric form of the functional equation
....... 186
8.2
0-estimates for
zeta
........................
187
8.3
Zeros of
zeta
............................ 189
8.4
A zerofree region for
zeta
..................... 192
8.5
An estimate of
ζ /ζ.........................
197
8.6
Estimation of
φ
........................... 199
8.7
The P.N.T. with a remainder term
................ 202
8.8
Estimation of
M
.......................... 208
8.9
The density of zeros in the critical strip
............. 210
8.10
An explicit formula for
φι
..................... 213
8.11
Notes
................................ 219
Chapter
9
Primes in Arithmetic Progressions
221
9.1
Residue characters
......................... 221
9.2
Group structure of the
coprirne
residue classes
......... 225
9.3
Existence of enough characters
.................. 226
xiv Contents
9.4
L
functions
............................. 228
9.5
Proof of Dirichlet s theorem
.................... 231
9.6
P.N.T. for arithmetic progressions
................ 233
9.7
Notes
................................ 236
Chapter
10
Applications of Characters
237
10.1
Integers generated by primes in residue classes
......... 237
10.2
Sums of squares
........................... 242
10.3
A measure of nonprincipality
................... 247
10.4
Quadratic excess
.......................... 250
10.5
Evaluation of Gaussian sums
................... 254
10.6
Notes
................................ 258
Chapter
11
Oscillation Theorems
261
11.1
Introduction
............................. 261
11.2
Approximate periodicity
...................... 262
11.3
The use of Landau s oscillation theorem
............. 267
11.4
A quantitative estimate
...................... 269
11.5
The use of many singularities
................... 272
11.5.1
Applications
........................ 277
11.6
Sign changes of
π(χ)
—li
χ.....................
278
11.7
The size of
Μ (χ)
/л/х .......................
280
11.7.1
Numerical calculations
................... 285
11.8
The error term in the divisor problem
.............. 286
11.9
Notes
................................ 287
Chapter
12
Sieves
289
12.1
Introduction
............................ 289
12.2
The sieve of Eratosthenes and Legendre
............. 291
12.3
Sieve setup
............................. 293
12.4
The Brun-Hooley sieve
...................... 297
12.5
The large sieve
........................... 302
12.6
An extremal
majorant
....................... 303
12.7
Proof of Theorem
12.9 ...................... 309
12.8
Notes
................................ 312
Chapter
13
Application of Sieves
313
13.1
A Brun-Hooley estimate of twin primes
............. 313
Contents xv
13.2 The Brun-Titchmarsh
inequality
................. 315
13.3
Primes represented by polynomials
............... 319
13.4
A uniform two residue sieve estimate
.............. 325
13.5
Twin primes and Goldbach s problem
.............. 331
13.6
A heuristic formula for twin primes
............... 334
13.7
Notes
................................ 337
Appendix A Results from Analysis and Algebra
339
A.I Properties of real functions
.................... 339
A.I.I Decomposition
....................... 339
A.I.
2
Riemann-Stieltjes integrals
................ 340
A.1.3 Integrators
......................... 342
A.2 The
Euler
gamma function
.................... 346
A.3
Poisson
summation formula
.................... 347
A.4 Basis theorem for finite abelian groups
.............. 349
Bibliography
353
Index of Names and Topics
355
Index of Symbols
359
This valuable book focuses on a collection of powerful
¡(methods of analysis that yield deep number-theoretical if
estimates. Particular attention is given to counting
functions of prime numbers and multiplicative arithmetic
functions. Both real variable ( elementary ) and complex
variable ( analytic ) methods are employed. The reader is
assumed to have knowledge of elementary number theory -f
(abstract algebra will also do) and real and complex analysis.
Specialized analytic techniques, including transform
Tauberian methods, are developed as needed.
|
| any_adam_object | 1 |
| author | Bateman, Paul T. 1919-2012 Diamond, Harold G. 1940- |
| author_GND | (DE-588)117707791 (DE-588)142256420 |
| author_facet | Bateman, Paul T. 1919-2012 Diamond, Harold G. 1940- |
| author_role | aut aut |
| author_sort | Bateman, Paul T. 1919-2012 |
| author_variant | p t b pt ptb h g d hg hgd |
| building | Verbundindex |
| bvnumber | BV035366881 |
| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)552165842 (DE-599)BVBBV035366881 |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV035366881 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T21:32:15Z |
| institution | BVB |
| isbn | 9812389385 9812560807 |
| language | English |
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| publishDate | 2004 |
| publishDateSearch | 2004 |
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| publisher | World Scientific |
| record_format | marc |
| series | Monographs in number theory |
| series2 | Monographs in number theory |
| spelling | Bateman, Paul T. 1919-2012 Verfasser (DE-588)117707791 aut Analytic number theory an introductory course Paul T. Bateman ; Harold G. Diamond 1. publ. Singapore [u.a.] World Scientific 2004 XV, 360 S. txt rdacontent n rdamedia nc rdacarrier Monographs in number theory 1 Hier auch später erschienene, unveränderte Nachdrucke Analytische Zahlentheorie Nombres, Théorie des Number theory Analytische Zahlentheorie (DE-588)4001870-2 gnd rswk-swf Analytische Zahlentheorie (DE-588)4001870-2 s DE-604 Diamond, Harold G. 1940- Verfasser (DE-588)142256420 aut Monographs in number theory 1 (DE-604)BV035341434 1 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017170833&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017170833&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Bateman, Paul T. 1919-2012 Diamond, Harold G. 1940- Analytic number theory an introductory course Monographs in number theory Analytische Zahlentheorie Nombres, Théorie des Number theory Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| subject_GND | (DE-588)4001870-2 |
| title | Analytic number theory an introductory course |
| title_auth | Analytic number theory an introductory course |
| title_exact_search | Analytic number theory an introductory course |
| title_full | Analytic number theory an introductory course Paul T. Bateman ; Harold G. Diamond |
| title_fullStr | Analytic number theory an introductory course Paul T. Bateman ; Harold G. Diamond |
| title_full_unstemmed | Analytic number theory an introductory course Paul T. Bateman ; Harold G. Diamond |
| title_short | Analytic number theory |
| title_sort | analytic number theory an introductory course |
| title_sub | an introductory course |
| topic | Analytische Zahlentheorie Nombres, Théorie des Number theory Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| topic_facet | Analytische Zahlentheorie Nombres, Théorie des Number theory |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017170833&sequence=000002&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=017170833&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035341434 |
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