The Lerch zeta-function:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer
2002
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 180-185) and index |
| Physical Description: | VIII, 189 S. |
| ISBN: | 1402010141 |
Staff View
MARC
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| 245 | 1 | 0 | |a The Lerch zeta-function |c by Antanas Laurinčikas and Ramūnas Garunkštis |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 2002 | |
| 300 | |a VIII, 189 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 650 | 0 | 7 | |a Zetafunktion |0 (DE-588)4190764-4 |2 gnd |9 rswk-swf |
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Record in the Search Index
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Contents
Preface vii
1. EULER GAMMA FUNCTION 1
1.1 Definition and Analytic Continuation 1
1.2 Representation by an Infinite Product 3
1.3 Functional Equation 6
1.4 Complementary Formula 7
1.5 Asymptotic Formulas 9
1.6 Hypergeometric Function 12
Notes 14
2. FUNCTIONAL EQUATION 17
2.1 Definition of the Lerch Zeta Function 17
2.2 Analytic Continuation 18
2.3 Functional Equation 21
2.4 Application of the Euler Maclaurin Formula 23
Notes 28
3. MOMENTS 31
3.1 Approximation of L(\,a, s) by a Finite Sum 31
3.2 Montgomery Vaughan Theorem 35
3.3 Mean Square of L(k, a, s) 36
3.4 Mean Square of L(X, a, s) with Respect to a 39
Notes 48
4. APPROXIMATE FUNCTIONAL EQUATION 53
4.1 Proof of the Approximate Functional Equation 53
v
vi Contents
4.2 Application of the Approximate Functional Equation to the
Mean Square of L (A., a, s) 59
Notes 69
5. STATISTICAL PROPERTIES 71
5.1 Limit Theorems on the Complex Plane 71
5.2 Limit Theorems in the Space of Analytic Functions 80
5.3 Joint Limit Theorems in the Space of Analytic Functions 97
5.4 Limit Theorems in the Space of Analytic Functions with
Rational a 102
Notes 108
6. UNIVERSALITY 111
6.1 Case of Transcendental a 111
6.2 Case of Rational a 111
6.3 Joint Universality of Lerch Zeta Functions 121
6.4 Effectivization Problem of the Universality Theorem 130
Notes 133
7. FUNCTIONAL INDEPENDENCE 137
7.1 The One Dimensional Case 137
7.2 Joint Functional Independence 140
Notes 141
8. DISTRIBUTION OF ZEROS 143
8.1 Zero Free Region on the Right 143
8.2 Zero Free Regions on the Left 144
8.3 Number of Nontrivial Zeros 152
8.4 Estimates of the Number of Nontrivial Zeros 156
8.5 Sums over Nontrivial Zeros 170
Notes 175
References 180
Notation 186
Subject Index 188 |
| adam_txt |
Contents
Preface vii
1. EULER GAMMA FUNCTION 1
1.1 Definition and Analytic Continuation 1
1.2 Representation by an Infinite Product 3
1.3 Functional Equation 6
1.4 Complementary Formula 7
1.5 Asymptotic Formulas 9
1.6 Hypergeometric Function 12
Notes 14
2. FUNCTIONAL EQUATION 17
2.1 Definition of the Lerch Zeta Function 17
2.2 Analytic Continuation 18
2.3 Functional Equation 21
2.4 Application of the Euler Maclaurin Formula 23
Notes 28
3. MOMENTS 31
3.1 Approximation of L(\,a, s) by a Finite Sum 31
3.2 Montgomery Vaughan Theorem 35
3.3 Mean Square of L(k, a, s) 36
3.4 Mean Square of L(X, a, s) with Respect to a 39
Notes 48
4. APPROXIMATE FUNCTIONAL EQUATION 53
4.1 Proof of the Approximate Functional Equation 53
v
vi Contents
4.2 Application of the Approximate Functional Equation to the
Mean Square of L (A., a, s) 59
Notes 69
5. STATISTICAL PROPERTIES 71
5.1 Limit Theorems on the Complex Plane 71
5.2 Limit Theorems in the Space of Analytic Functions 80
5.3 Joint Limit Theorems in the Space of Analytic Functions 97
5.4 Limit Theorems in the Space of Analytic Functions with
Rational a 102
Notes 108
6. UNIVERSALITY 111
6.1 Case of Transcendental a 111
6.2 Case of Rational a 111
6.3 Joint Universality of Lerch Zeta Functions 121
6.4 Effectivization Problem of the Universality Theorem 130
Notes 133
7. FUNCTIONAL INDEPENDENCE 137
7.1 The One Dimensional Case 137
7.2 Joint Functional Independence 140
Notes 141
8. DISTRIBUTION OF ZEROS 143
8.1 Zero Free Region on the Right 143
8.2 Zero Free Regions on the Left 144
8.3 Number of Nontrivial Zeros 152
8.4 Estimates of the Number of Nontrivial Zeros 156
8.5 Sums over Nontrivial Zeros 170
Notes 175
References 180
Notation 186
Subject Index 188 |
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| id | DE-604.BV021570788 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T14:38:14Z |
| indexdate | 2025-03-03T13:02:20Z |
| institution | BVB |
| isbn | 1402010141 |
| language | English |
| lccn | 2002040744 |
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| physical | VIII, 189 S. |
| publishDate | 2002 |
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| publisher | Kluwer |
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| spelling | Laurinčikas, Antanas Verfasser aut The Lerch zeta-function by Antanas Laurinčikas and Ramūnas Garunkštis Dordrecht [u.a.] Kluwer 2002 VIII, 189 S. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (p. 180-185) and index Euler products Functions, Zeta Zetafunktion (DE-588)4190764-4 gnd rswk-swf Zetafunktion (DE-588)4190764-4 s DE-604 Garunkštis, Ramūnas Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014786599&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Laurinčikas, Antanas The Lerch zeta-function Euler products Functions, Zeta Zetafunktion (DE-588)4190764-4 gnd |
| subject_GND | (DE-588)4190764-4 |
| title | The Lerch zeta-function |
| title_auth | The Lerch zeta-function |
| title_exact_search | The Lerch zeta-function |
| title_exact_search_txtP | The Lerch zeta-function |
| title_full | The Lerch zeta-function by Antanas Laurinčikas and Ramūnas Garunkštis |
| title_fullStr | The Lerch zeta-function by Antanas Laurinčikas and Ramūnas Garunkštis |
| title_full_unstemmed | The Lerch zeta-function by Antanas Laurinčikas and Ramūnas Garunkštis |
| title_short | The Lerch zeta-function |
| title_sort | the lerch zeta function |
| topic | Euler products Functions, Zeta Zetafunktion (DE-588)4190764-4 gnd |
| topic_facet | Euler products Functions, Zeta Zetafunktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014786599&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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