Introduction to analytic number theory:
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| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
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New York [u.a.]
Springer
[ca.] 2003
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| Ausgabe: | [Nachdr.] |
| Schriftenreihe: | Untergraduate texts in mathematics
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| Beschreibung: | XII, 338 S. graph. Darst. |
| ISBN: | 0387901639 |
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| 100 | 1 | |a Apostol, Tom M. |d 1923-2016 |e Verfasser |0 (DE-588)117707104 |4 aut | |
| 245 | 1 | 0 | |a Introduction to analytic number theory |
| 250 | |a [Nachdr.] | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c [ca.] 2003 | |
| 300 | |a XII, 338 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Untergraduate texts in mathematics | |
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Datensatz im Suchindex
| _version_ | 1804135496798961664 |
|---|---|
| adam_text | Contents
Historical Introduction
Chapter
1
The Fundamental Theorem of Arithmetic
1.1
Introduction
13
1.2
Divisibility
14
1.3
Greatest common divisor
14
1.4
Prime numbers
16
1.5
The fundamental theorem of arithmetic
1 7
1.6
The series of reciprocals of the primes
18
1.7
The Euclidean algorithm
19
1.8
The greatest common divisor of more than two numbers
20
Exercises for Chapter
1 21
Chapter
2
Arithmetical Functions and Dirichlet Multiplication
2.1
Introduction
24
2.2
The
Möbius
function
μ(η)
24
2.3
The
Euler totient
function
φ(η)
25
2.4
A relation connecting
φ
and
μ
26
2.5
A product formula for
φ{η)
27
2.6
The Dirichlet product of arithmetical functions
29
2.7
Dirichlet inverses and the
Möbius
inversion formula
30
2.8
The Mangoldt function A(n)
32
2.9
Multiplicative functions
33
2.10
Multiplicative functions and Dirichlet multiplication
35
2.11
The inverse of a completely multiplicative function
36
vii
2.12
Liou
ville s function
/.(и)
37
2.13
The divisor functions
σ,(«)
38
2.14
Generalized convolutions
39
2.15
Formal power series
41
2.16
The Bell series of an arithmetical function
42
2.17
Bell series and Dirichlet multiplication
44
2.18
Derivatives of arithmetical functions
45
2.19
The Selberg identity
46
Exercises for Chapter
2 46
Chapter
3
Averages of Arithmetical Functions
3.1
Introduction
52
3.2
The big oh notation. Asymptotic equality of functions
53
3.3
Euler s summation formula
54
3.4
Some elementary asymptotic formulas
55
3.5
The average order of d{n)
57
3.6
The average order of the divisor functions
ах{п)
60
3.7
The average order of
φ(η)
61
3.8
An application to the distribution of lattice points visible from the origin
62
3.9
The average order of
μ(η)
and of A{n)
64
3.10
The partial sums of a Dirichlet product
65
3.11
Applications to
μ(η)
and A(n)
66
3.12
Another identity for the partial sums of a Dirichlet product
69
Exercises for Chapter
3 70
Chapter
4
Some Elementary Theorems on the Distribution of Prime
Numbers
4.1
Introduction
74
4.2
Chebyshev s functions
ψ(χ)
and Ax)
75
4.3
Relations connecting 9(x) and
π{χ)
76
4.4
Some equivalent forms of the prime number theorem
79
4.5
Inequalities for
π(η)
and
р„
82
4.6
Shapiro s Tauberian theorem
#5
4.7
Applications of Shapiro s theorem
88
4.8
An asymptotic formula for the partial sums
]Грѕх
(1/p)
89
4.9
The partial sums of the
Möbius
function
91
4.10
Brief sketch of an elementary proof of the prime number theorem
98
4.11
Selberg s asymptotic formula
99
Exercises for Chapter
4 101
Chapter
5
Congruences
5.1
Definition and basic properties of congruences
106
5.2
Residue classes and complete residue systems
109
5.3
Linear congruences
110
viii
5.4
Reduced
residue
systems and the
Euler-Fermat
theorem
113
5.5
Polynomial congruences modulo p. Lagrange s theorem
114
5.6
Applications of
Lagrange s
theorem
115
5.7
Simultaneous linear congruences. The Chinese remainder theorem
117
5.8
Applications of the Chinese remainder theorem
118
5.9
Polynomial congruences with prime power moduli
120
5.10
The principle of cross-classification
123
5.11
A decomposition property of reduced residue systems
125
Exercises for Chapter
5 126
Chapter
6
Finite Abelian Groups and Their Characters
6.1
Definitions
129
6.2
Examples of groups and subgroups
130
6.3
Elementary properties of groups
130
6.4
Construction of subgroups
131
6.5
Characters of finite abelian groups
/33
6.6
The character group
/55
6.7
The orthogonality relations for characters
136
6.8
Dirichlet characters
í
37
6.9
Sums involving Dirichlet characters
140
6.10
The nonvanishing of L(l,
χ)
for real nonprincipal
χ
141
Exercises for Chapter
6 143
Chapter
7
Dirichlet s Theorem on Primes in Arithmetic Progressions
7.1
Introduction
146
7.2
Dirichlet s theorem for primes of the form An
— 1
and
4л
+ 1 147
7.3
The plan of the proof of Dirichlet s theorem
148
7.4
Proof of Lemma
7.4 150
7.5
Proof of Lemma
7.5 151
7.6
Proof of Lemma
7.6 152
1.1
Proof of Lemma
7.8 153
7.8
Proof of Lemma
7.7 153
7.9
Distribution of primes in arithmetic progressions
154
Exercises for Chapter
7 155
Chapter
8
Periodic Arithmetical Functions and Gauss Sums
8.1
Functions periodic modulo
к
157
8.2
Existence of finite Fourier series for periodic arithmetical functions
158
8.3
Ramanujan s sum and generalizations
160
8.4
Multiplicative properties of the sums sk(n)
162
8.5
Gauss sums associated with Dirichlet characters
165
8.6
Dirichlet characters with nonvanishing Gauss sums
166
8.7
Induced moduli and primitive characters
167
ix
8.8
Further properties of induced moduli
168
8.9
The conductor of a character
/ 71
8.10
Primitive characters and separable Gauss sums
171
8.11
The finite Fourier series of the Dirichlet characters
172
8.12
Pólya s
inequality for the partial sums of primitive characters
173
Exercises for Chapter
8 175
Chapter
9
Quadratic Residues and the Quadratic Reciprocity Law
9.1
Quadratic residues
178
9.2
Legendre s symbol and its properties
/ 79
9.3
Evaluation of
{ — 1
¡p) and
(2
1p)
181
9.4
Gauss lemma
182
9.5
The quadratic reciprocity law
185
9.6
Applications of the reciprocity law
186
9.7
The Jacobi symbol
¡87
9.8
Applications to Diophantine equations
190
9.9
Gauss sums and the quadratic reciprocity law
192
9.10
The reciprocity law for quadratic Gauss sums
195
9.11
Another proof of the quadratic reciprocity law
200
Exercises for Chapter
9 201
Chapter
10
Primitive Roots
10.1
The exponent of a number mod m. Primitive roots
204
10.2
Primitive roots and reduced residue systems
205
10.3
The nonexistence of primitive roots mod
2
for a
> 3 206
10.4
The existence of primitive roots mod
ρ
for odd primes
ρ
206
10.5
Primitive roots and quadratic residues
208
10.6
The existence of primitive roots mod p
208
10.7
The existence of primitive roots mod
2p* 210
10.8
The nonexistence of primitive roots in the remaining cases
211
10.9
The number of primitive roots mod
m
2/2
10.10
The index calculus
213
10.11
Primitive roots and Dirichlet characters
218
10.12
Real-valued Dirichlet characters mod p
220
10.13
Primitive Dirichlet characters mod p
221
Exercises for Chapter
10 222
Chapter
11
Dirichlet Series and
Euler
Products
11.1
Introduction
224
11.2
The half-plane of absolute convergence of a Dirichlet seiies
225
11.3
The function defined by a Dirichlet series
226
11.4
Multiplication
of Dirichlet series
228
11.5
Euler
products
230
11.6
The half-plane of convergence of a Dirichlet series
232
11.7
Analytic properties of Dirichlet series
234
11.8
Dirichlet series with
nonnegative
coefficients
236
11.9
Dirichlet series expressed as exponentials of Dirichlet series
238
11.10
Mean value formulas for Dirichlet series
240
11.11
An integral formula for the coefficients of a Dirichlet series
242
11.12
An integral formula for the partial sums of a Dirichlet series
243
Exercises for Chapter
11 246
Chapter
12
The Functions C(s) and L(s,
χ)
12.1
Introduction
249
12.2
Properties of the gamma function
250
12.3
Integral representation for the Hurwitz
zeta
function
251
12.4
A contour integral representation for the Hurwitz
zeta
function
253
12.5
The analytic continuation of the Hurwitz
zeta
function
254
12.6
Analytic continuation of
ζ{ς)
and L(s,
χ)
255
12.7
Hurwitz s formula for
f(s,
a)
256
12.8
The functional equation for the Riemann
zeta
function
259
12.9
A functional equation for the Hurwitz
zeta
function
261
12.10
The functional equation for L-functions
261
12.11
Evaluation of
{(-и, а)
264
12.12
Properties of Bernoulli numbers and Bernoulli polynomials
265
12.13
Formulas for L(0,
χ)
268
12.14
Approximation of
f (s,
α)
by finite sums
268
12.15
Inequalities for | C(s, a) |
270
12.16
Inequalities for |
f
(s)| and |L(s,
χ)
272
Exercises for Chapter
12 273
Chapter
13
Analytic Proof of the Prime Number Theorem
13.1
The plan of the proof
278
13.2
Lemmas
279
13.3
A contour integral representation for
ф^хУх2
283
13.4
Upper bounds for |f(s)| and |f (s)| near the line
σ
- 1 284
13.5
The nonvanishing of C(s) on the line
σ
= 1 286
13.6
Inequalities for |
í/[(s)¡
and |
ШШ
|
287
13.7
Completion of the proof of the prime number theorem
289
13.8
Zero-free regions for i(s)
291
13.9
The Riemann hypothesis
293
13.10
Application to the divisor function
294
13.11
Application to Euler s totient
297
13.12
Extension of
Pólya s
inequality for character sums
299
Exercises for Chapter
13 300
Chapter
14
Partitions
14.1
Introduction
304
14.2
Geometric representation of partitions
307
14.3
Generating functions for partitions
308
14.4
Euler s pentagonal-number theorem
311
14.5
Combinatorial proof of Euler s pentagonal-number theorem
313
14.6
Euler s recursion formula for p(n)
315
14.7
An upper bound for p(n)
316
14.8
Jacobi s triple product identity
318
14.9
Consequences of Jacobi s identity
321
14.10
Logarithmic differentiation of generating functions
322
14.11
The partition identities of Ramanujan
324
Exercises for Chapter
14 325
Bibliography
329
Index of Special Symbols
333
Index
335
Xli
|
| adam_txt |
Contents
Historical Introduction
Chapter
1
The Fundamental Theorem of Arithmetic
1.1
Introduction
13
1.2
Divisibility
14
1.3
Greatest common divisor
14
1.4
Prime numbers
16
1.5
The fundamental theorem of arithmetic
1 7
1.6
The series of reciprocals of the primes
18
1.7
The Euclidean algorithm
19
1.8
The greatest common divisor of more than two numbers
20
Exercises for Chapter
1 21
Chapter
2
Arithmetical Functions and Dirichlet Multiplication
2.1
Introduction
24
2.2
The
Möbius
function
μ(η)
24
2.3
The
Euler totient
function
φ(η)
25
2.4
A relation connecting
φ
and
μ
26
2.5
A product formula for
φ{η)
27
2.6
The Dirichlet product of arithmetical functions
29
2.7
Dirichlet inverses and the
Möbius
inversion formula
30
2.8
The Mangoldt function A(n)
32
2.9
Multiplicative functions
33
2.10
Multiplicative functions and Dirichlet multiplication
35
2.11
The inverse of a completely multiplicative function
36
vii
2.12
Liou
ville's function
/.(и)
37
2.13
The divisor functions
σ,(«)
38
2.14
Generalized convolutions
39
2.15
Formal power series
41
2.16
The Bell series of an arithmetical function
42
2.17
Bell series and Dirichlet multiplication
44
2.18
Derivatives of arithmetical functions
45
2.19
The Selberg identity
46
Exercises for Chapter
2 46
Chapter
3
Averages of Arithmetical Functions
3.1
Introduction
52
3.2
The big oh notation. Asymptotic equality of functions
53
3.3
Euler's summation formula
54
3.4
Some elementary asymptotic formulas
55
3.5
The average order of d{n)
57
3.6
The average order of the divisor functions
ах{п)
60
3.7
The average order of
φ(η)
61
3.8
An application to the distribution of lattice points visible from the origin
62
3.9
The average order of
μ(η)
and of A{n)
64
3.10
The partial sums of a Dirichlet product
65
3.11
Applications to
μ(η)
and A(n)
66
3.12
Another identity for the partial sums of a Dirichlet product
69
Exercises for Chapter
3 70
Chapter
4
Some Elementary Theorems on the Distribution of Prime
Numbers
4.1
Introduction
74
4.2
Chebyshev's functions
ψ(χ)
and 'Ax)
75
4.3
Relations connecting 9(x) and
π{χ)
76
4.4
Some equivalent forms of the prime number theorem
79
4.5
Inequalities for
π(η)
and
р„
82
4.6
Shapiro's Tauberian theorem
#5
4.7
Applications of Shapiro's theorem
88
4.8
An asymptotic formula for the partial sums
]Грѕх
(1/p)
89
4.9
The partial sums of the
Möbius
function
91
4.10
Brief sketch of an elementary proof of the prime number theorem
98
4.11
Selberg's asymptotic formula
99
Exercises for Chapter
4 101
Chapter
5
Congruences
5.1
Definition and basic properties of congruences
106
5.2
Residue classes and complete residue systems
109
5.3
Linear congruences
110
viii
5.4
Reduced
residue
systems and the
Euler-Fermat
theorem
113
5.5
Polynomial congruences modulo p. Lagrange's theorem
114
5.6
Applications of
Lagrange 's
theorem
115
5.7
Simultaneous linear congruences. The Chinese remainder theorem
117
5.8
Applications of the Chinese remainder theorem
118
5.9
Polynomial congruences with prime power moduli
120
5.10
The principle of cross-classification
123
5.11
A decomposition property of reduced residue systems
125
Exercises for Chapter
5 126
Chapter
6
Finite Abelian Groups and Their Characters
6.1
Definitions
129
6.2
Examples of groups and subgroups
130
6.3
Elementary properties of groups
130
6.4
Construction of subgroups
131
6.5
Characters of finite abelian groups
/33
6.6
The character group
/55
6.7
The orthogonality relations for characters
136
6.8
Dirichlet characters
í
37
6.9
Sums involving Dirichlet characters
140
6.10
The nonvanishing of L(l,
χ)
for real nonprincipal
χ
141
Exercises for Chapter
6 143
Chapter
7
Dirichlet's Theorem on Primes in Arithmetic Progressions
7.1
Introduction
146
7.2
Dirichlet's theorem for primes of the form An
— 1
and
4л
+ 1 147
7.3
The plan of the proof of Dirichlet's theorem
148
7.4
Proof of Lemma
7.4 150
7.5
Proof of Lemma
7.5 151
7.6
Proof of Lemma
7.6 152
1.1
Proof of Lemma
7.8 153
7.8
Proof of Lemma
7.7 153
7.9
Distribution of primes in arithmetic progressions
154
Exercises for Chapter
7 155
Chapter
8
Periodic Arithmetical Functions and Gauss Sums
8.1
Functions periodic modulo
к
157
8.2
Existence of finite Fourier series for periodic arithmetical functions
158
8.3
Ramanujan's sum and generalizations
160
8.4
Multiplicative properties of the sums sk(n)
162
8.5
Gauss sums associated with Dirichlet characters
165
8.6
Dirichlet characters with nonvanishing Gauss sums
166
8.7
Induced moduli and primitive characters
167
ix
8.8
Further properties of induced moduli
168
8.9
The conductor of a character
/ 71
8.10
Primitive characters and separable Gauss sums
171
8.11
The finite Fourier series of the Dirichlet characters
172
8.12
Pólya's
inequality for the partial sums of primitive characters
173
Exercises for Chapter
8 175
Chapter
9
Quadratic Residues and the Quadratic Reciprocity Law
9.1
Quadratic residues
178
9.2
Legendre's symbol and its properties
/ 79
9.3
Evaluation of
{ — 1
¡p) and
(2
1p)
181
9.4
Gauss' lemma
182
9.5
The quadratic reciprocity law
185
9.6
Applications of the reciprocity law
186
9.7
The Jacobi symbol
¡87
9.8
Applications to Diophantine equations
190
9.9
Gauss sums and the quadratic reciprocity law
192
9.10
The reciprocity law for quadratic Gauss sums
195
9.11
Another proof of the quadratic reciprocity law
200
Exercises for Chapter
9 201
Chapter
10
Primitive Roots
10.1
The exponent of a number mod m. Primitive roots
204
10.2
Primitive roots and reduced residue systems
205
10.3
The nonexistence of primitive roots mod
2"
for a
> 3 206
10.4
The existence of primitive roots mod
ρ
for odd primes
ρ
206
10.5
Primitive roots and quadratic residues
208
10.6
The existence of primitive roots mod p"
208
10.7
The existence of primitive roots mod
2p* 210
10.8
The nonexistence of primitive roots in the remaining cases
211
10.9
The number of primitive roots mod
m
2/2
10.10
The index calculus
213
10.11
Primitive roots and Dirichlet characters
218
10.12
Real-valued Dirichlet characters mod p"
220
10.13
Primitive Dirichlet characters mod p"
221
Exercises for Chapter
10 222
Chapter
11
Dirichlet Series and
Euler
Products
11.1
Introduction
224
11.2
The half-plane of absolute convergence of a Dirichlet seiies
225
11.3
The function defined by a Dirichlet series
226
11.4
Multiplication
of Dirichlet series
228
11.5
Euler
products
230
11.6
The half-plane of convergence of a Dirichlet series
232
11.7
Analytic properties of Dirichlet series
234
11.8
Dirichlet series with
nonnegative
coefficients
236
11.9
Dirichlet series expressed as exponentials of Dirichlet series
238
11.10
Mean value formulas for Dirichlet series
240
11.11
An integral formula for the coefficients of a Dirichlet series
242
11.12
An integral formula for the partial sums of a Dirichlet series
243
Exercises for Chapter
11 246
Chapter
12
The Functions C(s) and L(s,
χ)
12.1
Introduction
249
12.2
Properties of the gamma function
250
12.3
Integral representation for the Hurwitz
zeta
function
251
12.4
A contour integral representation for the Hurwitz
zeta
function
253
12.5
The analytic continuation of the Hurwitz
zeta
function
254
12.6
Analytic continuation of
ζ{ς)
and L(s,
χ)
255
12.7
Hurwitz's formula for
f(s,
a)
256
12.8
The functional equation for the Riemann
zeta
function
259
12.9
A functional equation for the Hurwitz
zeta
function
261
12.10
The functional equation for L-functions
261
12.11
Evaluation of
{(-и, а)
264
12.12
Properties of Bernoulli numbers and Bernoulli polynomials
265
12.13
Formulas for L(0,
χ)
268
12.14
Approximation of
f (s,
α)
by finite sums
268
12.15
Inequalities for | C(s, a) |
270
12.16
Inequalities for |
f
(s)| and |L(s,
χ)\
272
Exercises for Chapter
12 273
Chapter
13
Analytic Proof of the Prime Number Theorem
13.1
The plan of the proof
278
13.2
Lemmas
279
13.3
A contour integral representation for
ф^хУх2
283
13.4
Upper bounds for |f(s)| and |f'(s)| near the line
σ
- 1 284
13.5
The nonvanishing of C(s) on the line
σ
= 1 286
13.6
Inequalities for |
í/[(s)¡
and |
ШШ
|
287
13.7
Completion of the proof of the prime number theorem
289
13.8
Zero-free regions for i(s)
291
13.9
The Riemann hypothesis
293
13.10
Application to the divisor function
294
13.11
Application to Euler's totient
297
13.12
Extension of
Pólya's
inequality for character sums
299
Exercises for Chapter
13 300
Chapter
14
Partitions
14.1
Introduction
304
14.2
Geometric representation of partitions
307
14.3
Generating functions for partitions
308
14.4
Euler's pentagonal-number theorem
311
14.5
Combinatorial proof of Euler's pentagonal-number theorem
313
14.6
Euler's recursion formula for p(n)
315
14.7
An upper bound for p(n)
316
14.8
Jacobi's triple product identity
318
14.9
Consequences of Jacobi's identity
321
14.10
Logarithmic differentiation of generating functions
322
14.11
The partition identities of Ramanujan
324
Exercises for Chapter
14 325
Bibliography
329
Index of Special Symbols
333
Index
335
Xli |
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| author | Apostol, Tom M. 1923-2016 |
| author_GND | (DE-588)117707104 |
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| building | Verbundindex |
| bvnumber | BV021677506 |
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| ctrlnum | (OCoLC)634647652 (DE-599)BVBBV021677506 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | [Nachdr.] |
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| physical | XII, 338 S. graph. Darst. |
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| spelling | Apostol, Tom M. 1923-2016 Verfasser (DE-588)117707104 aut Introduction to analytic number theory [Nachdr.] New York [u.a.] Springer [ca.] 2003 XII, 338 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Untergraduate texts in mathematics Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Analytische Zahlentheorie (DE-588)4001870-2 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content 2\p (DE-588)4151278-9 Einführung gnd-content Zahlentheorie (DE-588)4067277-3 s DE-604 Analytische Zahlentheorie (DE-588)4001870-2 s 3\p DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014891771&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Apostol, Tom M. 1923-2016 Introduction to analytic number theory Zahlentheorie (DE-588)4067277-3 gnd Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| subject_GND | (DE-588)4067277-3 (DE-588)4001870-2 (DE-588)4123623-3 (DE-588)4151278-9 |
| title | Introduction to analytic number theory |
| title_auth | Introduction to analytic number theory |
| title_exact_search | Introduction to analytic number theory |
| title_exact_search_txtP | Introduction to analytic number theory |
| title_full | Introduction to analytic number theory |
| title_fullStr | Introduction to analytic number theory |
| title_full_unstemmed | Introduction to analytic number theory |
| title_short | Introduction to analytic number theory |
| title_sort | introduction to analytic number theory |
| topic | Zahlentheorie (DE-588)4067277-3 gnd Analytische Zahlentheorie (DE-588)4001870-2 gnd |
| topic_facet | Zahlentheorie Analytische Zahlentheorie Lehrbuch Einführung |
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