Screen LibraryContributions to the theory of zeta-functions : the modular relation supremacy /
This volume provides a systematic survey of almost all the equivalent assertions to the functional equations - zeta symmetry - which zeta-functions satisfy, thus streamlining previously published results on zeta-functions. The equivalent relations are given in the form of modular relations in Fox H-...
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| Main Authors: | |
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| Corporate Authors: | |
| Group Author: | |
| Published: |
World Scientific Pub. Co.,
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| Publisher Address: | Singapore ; Hackensack, N.J. : |
| Publication Dates: | 2015. |
| Literature type: | eBook |
| Language: | English |
| Series: |
Series on number theory and its applications ;
vol. 10 |
| Subjects: | |
| Online Access: |
http://www.worldscientific.com/worldscibooks/10.1142/8711#t=toc |
| Summary: |
This volume provides a systematic survey of almost all the equivalent assertions to the functional equations - zeta symmetry - which zeta-functions satisfy, thus streamlining previously published results on zeta-functions. The equivalent relations are given in the form of modular relations in Fox H-function series, which at present include all that have been considered as candidates for ingredients of a series. The results are presented in a clear and simple manner for readers to readily apply without much knowledge of zeta-functions. This volume aims to keep a record of the 150-year-old heritage starting from Riemann on zeta-functions, which are ubiquitous in all mathematical sciences, wherever there is a notion of the norm. It provides almost all possible equivalent relations to the zeta-functions without requiring a reader's deep knowledge on their definitions. This can be an ideal reference book for those studying zeta-functions. |
| Carrier Form: | 1 online resource (xii,303pages) : illustrations. |
| Bibliography: | Includes bibliographical references (pages 279-300) and index. |
| ISBN: | 9789814449625 |
| Index Number: | QA351 |
| CLC: | O156.4 |
| Contents: | 1. Prelude. 1.1. Introduction. 1.2. Eternal return or every 50 years. 1.3. The theta-transformation formula. 1.4. Summation formulas -- 2. Grocery of special functions. 2.1. Formulas for the gamma function and their use. 2.2. Zeta-functions. 2.3. Bessel functions. 2.4. [symbol]-functions. 2.5. Generalized hypergeometric functions. 2.6. Fox H-functions. 2.7. Formulas for Fox and Meijer functions. 2.8. Special cases of G-functions -- 3. Unprocessed modular relations. 3.1. The [symbol] formula. 3.2. Dedekind zeta-function I. 3.3. Transformation formulas for Lambert series. 3.4. Koshlyakov's method [KoshI]. 3.5. Koshlyakov's functions -- 4. Fourier-Bessel expansion [symbols]. 4.1. Introduction. 4.2. Stark's method. 4.3. The main formula for modular relations. 4.4. Dedekind zeta-function III. 4.5. Elucidation of Koshlyakov's result in the real quadratic case. 4.6. Koshlyakov's K-series. 4.7. The Fourier-Bessel expansion [symbols]. 4.8. Bochner-Chandrasekharan and Narasimhan formula -- 5. The Ewald expansion or the incomplete gamma series. 5.1. Ewald expansion for zeta-functions with a single gamma factor. 5.2. Atkinson-Berndt Abel mean -- 6. The Riesz sums. 6.1. Various modular relations. 6.2. Modular relations in integral form. 6.3. Integrated modular relations -- 7. The general modular relation. 7.1. Definitions. 7.2. Assumptions. 7.3. Theorem. 7.4. The Main Formula (basic version) -- 8. The Hecke type zeta-functions. 8.1. Statement of the formula. 8.2. The Riesz sums or the first J-Bessel expansion: [symbols]. 8.3. The partial sum formula: [symbols]. 8.4. The Fourier-Bessel expansion: [symbols]. 8.5. The Ewald expansion: [symbols]. 8.6. The Bochner-Chandrasekharan formula: [symbols]. 8.7. The [symbols] formula. 8.8. The second J-Bessel expansion: [symbols]. 8.9. The [symbols] formula. 8.10. The second K-Bessel expansion: [symbols]. 8.11. The [symbols] formula. 8.12. The [symbols] formula. 8.13. The [symbols] formula. 8.14. The [symbols] formula. 8.15. The [symbols] formula. 8.16. The [symbols] formula -- 9. The product of zeta-functions. 9.1 The product of zeta-functions. 9.2. Powers of zeta-functions -- 10. Miscellany. 10.1 Future projects. 10.2. Quellenangaben. |