Problems in analytic number theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
2008
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Graduate texts in mathematics
206 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXI, 502 S. |
| ISBN: | 9780387723495 |
Internformat
MARC
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| 100 | 1 | |a Murty, Maruti Ram |d 1953- |e Verfasser |0 (DE-588)120837307 |4 aut | |
| 245 | 1 | 0 | |a Problems in analytic number theory |c M. Ram Murty |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2008 | |
| 300 | |a XXI, 502 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 206 | |
| 490 | 0 | |a Readings in mathematics | |
| 650 | 4 | |a Number theory | |
| 650 | 0 | 7 | |a Analytische Zahlentheorie |0 (DE-588)4001870-2 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016538619 | ||
Datensatz im Suchindex
| _version_ | 1804137714809831424 |
|---|---|
| adam_text | Contents
Preface
to the Second Edition
vii
Acknowledgments for the Second Edition
ix
Preface to the First Edition
xi
Acknowledgments for the First Edition
xv
I Problems
1
1
Arithmetic Functions
3
1.1
The
Möbius
Inversion Formula and Applications
. . 4
1.2
Formal Dirichlet Series
................. 7
1.3
Orders of Some Arithmetical Functions
....... 9
1.4
Average Orders of Arithmetical Functions
...... 10
1.5
Supplementary Problems
............... 11
2
Primes in Arithmetic Progressions
17
2.1
Summation Techniques
................. 17
2.2
Characters mod
g
.................... 22
2.3
Dirichleťs
Theorem
................... 24
xviii Contents
2.4
Dirichleťs Hyperbola
Method
............. 27
2.5
Supplementary Problems
............... 29
3
The Prime Number Theorem
35
3.1
Chebyshev s Theorem
................. 36
3.2
Nonvanishing of Dirichlet Series on Re(s)
= 1 . . . 39
3.3
The
Ikehara
-
Wiener Theorem
............ 42
3.4
Supplementary Problems
............... 48
4
The Method of Contour Integration
53
4.1
Some Basic Integrals
.................. 53
4.2
The Prime Number Theorem
............. 57
4.3
Further Examples
.................... 62
4.4
Supplementary Problems
............... 65
5
Functional Equations
69
5.1
Poisson s Summation Formula
............ 69
5.2
The Riemann
Zeta
Function
.............. 72
5.3
Gauss Sums
....................... 75
5.4
Dirichlet L-functions
.................. 76
5.5
Supplementary Problems
............... 79
6
Hadamard
Products
85
6.1
Jensen s Theorem
.................... 85
6.2
Entire Functions of Order
1 .............. 88
6.3
The Gamma Function
.................. 91
6.4
Infinite Products for £(s) and £(s,%)
......... 93
6.5
Zero-Free Regions for
ζ(β)
and L(s,
χ)
........ 94
6.6
Supplementary Problems
............... 99
7
Explicit Formulas
101
7.1
Counting Zeros
..................... 101
7.2
Explicit Formula for
φ
(ж) ...............
104
7.3
Weil s Explicit Formula
................. 107
7.4
Supplementary Problems
............... 110
8
The Selberg Class
115
8.1
The
Phragmén
-
Lindelof
Theorem
.......... 116
8.2
Basic Properties
..................... 118
8.3
Selberg s Conjectures
.................. 123
Contents xix
8.4
Supplementary
Problems ............... 125
9
Sieve Methods
127
9.1
The Sieve of Eratosthenes
............... 127
9.2
Brun s Elementary Sieve
................ 133
9.3
Selberg s Sieve
...................... 138
9.4
Supplementary Problems
............... 144
10
/7-adic Methods
147
10.1
Ostrowski s Theorem
.................. 147
10.2
Hensel s Lemma
..................... 155
10.3
p-adic Interpolation
................... 159
10.4
Thep-adicZeta-Function
................ 165
10.5
Supplementary Problems
............... 168
11
Equidistribution
171
11.1
Uniform distribution modulo
1............ 171
11.2
Normal numbers
.................... 177
11.3
Asymptotic distribution functions mod
1...... 180
11.4
Discrepancy
....................... 182
11.5
Equidistribution and L-functions
........... 189
11.6
Supplementary Problems
............... 193
II Solutions
197
1
Arithmetic Functions
199
1.1
The
Möbius
Inversion Formula and Applications
. . 199
1.2
Formal Dirichlet Series
................. 208
1.3
Orders of Some Arithmetical Functions
....... 212
1.4
Average Orders of Arithmetical Functions
...... 215
1.5
Supplementary Problems
............... 220
2
Primes in Arithmetic Progressions
237
2.1
Characters mod
g
.................... 237
2.2
Dirichleťs
Theorem
................... 246
2.3
Dirichleťs
Hyperbola Method
............. 251
2.4
Supplementary Problems
............... 257
3
The Prime Number Theorem
273
3.1
Chebyshev s Theorem
................. 273
xx Contents
3.2 Nonvanishing
of Dirichlet Series on Re(s)
= 1 ... 280
3.3
The
Ikehara
-
Wiener Theorem
............ 288
3.4
Supplementary Problems
............... 292
4
The Method of Contour Integration
305
4.1
Some Basic Integrals
.................. 305
4.2
The Prime Number Theorem
............. 311
4.3
Further Examples
.................... 314
4.4
Supplementary Problems
............... 317
5
Functional Equations
329
5.1
Poisson s Summation Formula
............ 329
5.2
The Riemann
Zeta
Function
.............. 332
5.3
Gauss Sums
....................... 333
5.4
Dirichlet L-functions
.................. 335
5.5
Supplementary Problems
............... 338
6
Hadamard
Products
357
6.1
Jensen s theorem
.................... 357
6.2
The Gamma Function
.................. 358
6.3
infinite Products for £(s) and £(s,x)
......... 369
6.4
Zero-Free Regions for £(s) and L(s,x)
........ 374
6.5
Supplementary Problems
............... 379
7
Explicit Formulas
385
7.1
Counting Zeros
..................... 385
7.2
Explicit Formula for
φ (χ) ...............
388
7.3
Supplementary Problems
............... 393
8
The Selberg Class
403
8.1
The
Phragmén
-
Lindelof
Theorem
.......... 403
8.2
Basic Properties
..................... 404
8.3
Selberg s Conjectures
.................. 411
8.4
Supplementary Problems
............... 417
9
Sieve Methods
423
9.1
The Sieve of Eratosthenes
............... 423
9.2
Brun s Elementary Sieve
................ 429
9.3
Selberg s Sieve
...................... 432
9.4
Supplementary Problems
............... 439
Contents xxi
10
p-adic Methods
449
10.1
Ostrowski s Theorem
.................. 449
10.2
Henseľs
Lemma
..................... 454
10.3
p-adic Interpolation
................... 456
10.4
The p-adic ¿-Function
.................. 463
10.5
Supplementary Problems
............... 468
11
Equidistribution
475
11.1
Uniform distribution modulo
1............ 475
11.2
Normal numbers
.................... 483
11.3
Asymptotic distribution functions mod
1...... 484
11.4
Discrepancy
....................... 485
11.5
Equidistribution and L-functions
........... 488
11.6
Supplementary Problems
............... 490
References
497
Index
499
|
| adam_txt |
Contents
Preface
to the Second Edition
vii
Acknowledgments for the Second Edition
ix
Preface to the First Edition
xi
Acknowledgments for the First Edition
xv
I Problems
1
1
Arithmetic Functions
3
1.1
The
Möbius
Inversion Formula and Applications
. . 4
1.2
Formal Dirichlet Series
. 7
1.3
Orders of Some Arithmetical Functions
. 9
1.4
Average Orders of Arithmetical Functions
. 10
1.5
Supplementary Problems
. 11
2
Primes in Arithmetic Progressions
17
2.1
Summation Techniques
. 17
2.2
Characters mod
g
. 22
2.3
Dirichleťs
Theorem
. 24
xviii Contents
2.4
Dirichleťs Hyperbola
Method
. 27
2.5
Supplementary Problems
. 29
3
The Prime Number Theorem
35
3.1
Chebyshev's Theorem
. 36
3.2
Nonvanishing of Dirichlet Series on Re(s)
= 1 . . . 39
3.3
The
Ikehara
-
Wiener Theorem
. 42
3.4
Supplementary Problems
. 48
4
The Method of Contour Integration
53
4.1
Some Basic Integrals
. 53
4.2
The Prime Number Theorem
. 57
4.3
Further Examples
. 62
4.4
Supplementary Problems
. 65
5
Functional Equations
69
5.1
Poisson's Summation Formula
. 69
5.2
The Riemann
Zeta
Function
. 72
5.3
Gauss Sums
. 75
5.4
Dirichlet L-functions
. 76
5.5
Supplementary Problems
. 79
6
Hadamard
Products
85
6.1
Jensen's Theorem
. 85
6.2
Entire Functions of Order
1 . 88
6.3
The Gamma Function
. 91
6.4
Infinite Products for £(s) and £(s,%)
. 93
6.5
Zero-Free Regions for
ζ(β)
and L(s,
χ)
. 94
6.6
Supplementary Problems
. 99
7
Explicit Formulas
101
7.1
Counting Zeros
. 101
7.2
Explicit Formula for
φ
(ж) .
104
7.3
Weil's Explicit Formula
. 107
7.4
Supplementary Problems
. 110
8
The Selberg Class
115
8.1
The
Phragmén
-
Lindelof
Theorem
. 116
8.2
Basic Properties
. 118
8.3
Selberg's Conjectures
. 123
Contents xix
8.4
Supplementary
Problems . 125
9
Sieve Methods
127
9.1
The Sieve of Eratosthenes
. 127
9.2
Brun's Elementary Sieve
. 133
9.3
Selberg's Sieve
. 138
9.4
Supplementary Problems
. 144
10
/7-adic Methods
147
10.1
Ostrowski's Theorem
. 147
10.2
Hensel's Lemma
. 155
10.3
p-adic Interpolation
. 159
10.4
Thep-adicZeta-Function
. 165
10.5
Supplementary Problems
. 168
11
Equidistribution
171
11.1
Uniform distribution modulo
1. 171
11.2
Normal numbers
. 177
11.3
Asymptotic distribution functions mod
1. 180
11.4
Discrepancy
. 182
11.5
Equidistribution and L-functions
. 189
11.6
Supplementary Problems
. 193
II Solutions
197
1
Arithmetic Functions
199
1.1
The
Möbius
Inversion Formula and Applications
. . 199
1.2
Formal Dirichlet Series
. 208
1.3
Orders of Some Arithmetical Functions
. 212
1.4
Average Orders of Arithmetical Functions
. 215
1.5
Supplementary Problems
. 220
2
Primes in Arithmetic Progressions
237
2.1
Characters mod
g
. 237
2.2
Dirichleťs
Theorem
. 246
2.3
Dirichleťs
Hyperbola Method
. 251
2.4
Supplementary Problems
. 257
3
The Prime Number Theorem
273
3.1
Chebyshev's Theorem
. 273
xx Contents
3.2 Nonvanishing
of Dirichlet Series on Re(s)
= 1 . 280
3.3
The
Ikehara
-
Wiener Theorem
. 288
3.4
Supplementary Problems
. 292
4
The Method of Contour Integration
305
4.1
Some Basic Integrals
. 305
4.2
The Prime Number Theorem
. 311
4.3
Further Examples
. 314
4.4
Supplementary Problems
. 317
5
Functional Equations
329
5.1
Poisson's Summation Formula
. 329
5.2
The Riemann
Zeta
Function
. 332
5.3
Gauss Sums
. 333
5.4
Dirichlet L-functions
. 335
5.5
Supplementary Problems
. 338
6
Hadamard
Products
357
6.1
Jensen's theorem
. 357
6.2
The Gamma Function
. 358
6.3
infinite Products for £(s) and £(s,x)
. 369
6.4
Zero-Free Regions for £(s) and L(s,x)
. 374
6.5
Supplementary Problems
. 379
7
Explicit Formulas
385
7.1
Counting Zeros
. 385
7.2
Explicit Formula for
φ (χ) .
388
7.3
Supplementary Problems
. 393
8
The Selberg Class
403
8.1
The
Phragmén
-
Lindelof
Theorem
. 403
8.2
Basic Properties
. 404
8.3
Selberg's Conjectures
. 411
8.4
Supplementary Problems
. 417
9
Sieve Methods
423
9.1
The Sieve of Eratosthenes
. 423
9.2
Brun's Elementary Sieve
. 429
9.3
Selberg's Sieve
. 432
9.4
Supplementary Problems
. 439
Contents xxi
10
p-adic Methods
449
10.1
Ostrowski's Theorem
. 449
10.2
Henseľs
Lemma
. 454
10.3
p-adic Interpolation
. 456
10.4
The p-adic ¿-Function
. 463
10.5
Supplementary Problems
. 468
11
Equidistribution
475
11.1
Uniform distribution modulo
1. 475
11.2
Normal numbers
. 483
11.3
Asymptotic distribution functions mod
1. 484
11.4
Discrepancy
. 485
11.5
Equidistribution and L-functions
. 488
11.6
Supplementary Problems
. 490
References
497
Index
499 |
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| id | DE-604.BV023355070 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T21:06:42Z |
| indexdate | 2024-07-09T21:16:41Z |
| institution | BVB |
| isbn | 9780387723495 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016538619 |
| oclc_num | 255689214 |
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| owner_facet | DE-355 DE-BY-UBR DE-83 DE-11 DE-20 |
| physical | XXI, 502 S. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
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| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics Readings in mathematics |
| spelling | Murty, Maruti Ram 1953- Verfasser (DE-588)120837307 aut Problems in analytic number theory M. Ram Murty 2. ed. New York [u.a.] Springer 2008 XXI, 502 S. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 206 Readings in mathematics Number theory Analytische Zahlentheorie (DE-588)4001870-2 gnd rswk-swf Analytische Zahlentheorie (DE-588)4001870-2 s DE-604 Graduate texts in mathematics 206 (DE-604)BV000000067 206 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016538619&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Murty, Maruti Ram 1953- Problems in analytic number theory Graduate texts in mathematics Number theory Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| subject_GND | (DE-588)4001870-2 |
| title | Problems in analytic number theory |
| title_auth | Problems in analytic number theory |
| title_exact_search | Problems in analytic number theory |
| title_exact_search_txtP | Problems in analytic number theory |
| title_full | Problems in analytic number theory M. Ram Murty |
| title_fullStr | Problems in analytic number theory M. Ram Murty |
| title_full_unstemmed | Problems in analytic number theory M. Ram Murty |
| title_short | Problems in analytic number theory |
| title_sort | problems in analytic number theory |
| topic | Number theory Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| topic_facet | Number theory Analytische Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016538619&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT murtymarutiram problemsinanalyticnumbertheory |