Arithmetic Functions and Integer Products:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer New York
1985
|
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
272 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | Every positive integer m has a product representation of the form where v, k and the ni are positive integers, and each Ei = ± I. A value can be given for v which is uniform in the m. A representation can be computed so that no ni exceeds a certain fixed power of 2m, and the number k of terms needed does not exceed a fixed power of log 2m. Consider next the collection of finite probability spaces whose associated measures assume only rational values. Let hex) be a real-valued function which measures the information in an event, depending only upon the probability x with which that event occurs. Assuming hex) to be nonnegative, and to satisfy certain standard properties, it must have the form -A(x log x + (I - x) 10g(I -x». Except for a renormalization this is the well-known function of Shannon. What do these results have in common? They both apply the theory of arithmetic functions. The two widest classes of arithmetic functions are the real-valued additive and the complex-valued multiplicative functions. Beginning in the thirties of this century, the work of Erdos, Kac, Kubilius, Turan and others gave a discipline to the study of the general value distribution of arithmetic functions by the introduction of ideas, methods and results from the theory of Probability. I gave an account of the resulting extensive and still developing branch of Number Theory in volumes 239/240 of this series, under the title Probabilistic Number Theory |
| Beschreibung: | 1 Online-Ressource (461p) |
| ISBN: | 9781461385486 9781461385509 |
| DOI: | 10.1007/978-1-4613-8548-6 |
Internformat
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| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |v 272 | |
| 500 | |a Every positive integer m has a product representation of the form where v, k and the ni are positive integers, and each Ei = ± I. A value can be given for v which is uniform in the m. A representation can be computed so that no ni exceeds a certain fixed power of 2m, and the number k of terms needed does not exceed a fixed power of log 2m. Consider next the collection of finite probability spaces whose associated measures assume only rational values. Let hex) be a real-valued function which measures the information in an event, depending only upon the probability x with which that event occurs. Assuming hex) to be nonnegative, and to satisfy certain standard properties, it must have the form -A(x log x + (I - x) 10g(I -x». Except for a renormalization this is the well-known function of Shannon. What do these results have in common? They both apply the theory of arithmetic functions. The two widest classes of arithmetic functions are the real-valued additive and the complex-valued multiplicative functions. Beginning in the thirties of this century, the work of Erdos, Kac, Kubilius, Turan and others gave a discipline to the study of the general value distribution of arithmetic functions by the introduction of ideas, methods and results from the theory of Probability. I gave an account of the resulting extensive and still developing branch of Number Theory in volumes 239/240 of this series, under the title Probabilistic Number Theory | ||
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Datensatz im Suchindex
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| dewey-search | 512.7 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-1-4613-8548-6 |
| format | Electronic eBook |
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| indexdate | 2024-07-20T06:38:47Z |
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| isbn | 9781461385486 9781461385509 |
| language | English |
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| series | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |
| series2 | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |
| spelling | Elliott, P. D. T. A. Verfasser aut Arithmetic Functions and Integer Products by P. D. T. A. Elliott New York, NY Springer New York 1985 1 Online-Ressource (461p) txt rdacontent c rdamedia cr rdacarrier Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 272 Every positive integer m has a product representation of the form where v, k and the ni are positive integers, and each Ei = ± I. A value can be given for v which is uniform in the m. A representation can be computed so that no ni exceeds a certain fixed power of 2m, and the number k of terms needed does not exceed a fixed power of log 2m. Consider next the collection of finite probability spaces whose associated measures assume only rational values. Let hex) be a real-valued function which measures the information in an event, depending only upon the probability x with which that event occurs. Assuming hex) to be nonnegative, and to satisfy certain standard properties, it must have the form -A(x log x + (I - x) 10g(I -x». Except for a renormalization this is the well-known function of Shannon. What do these results have in common? They both apply the theory of arithmetic functions. The two widest classes of arithmetic functions are the real-valued additive and the complex-valued multiplicative functions. Beginning in the thirties of this century, the work of Erdos, Kac, Kubilius, Turan and others gave a discipline to the study of the general value distribution of arithmetic functions by the introduction of ideas, methods and results from the theory of Probability. I gave an account of the resulting extensive and still developing branch of Number Theory in volumes 239/240 of this series, under the title Probabilistic Number Theory Mathematics Number theory Number Theory Mathematik Arithmetische Funktion (DE-588)4368429-4 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Darstellung (DE-588)4200624-7 gnd rswk-swf Natürliche Zahl (DE-588)4041357-3 gnd rswk-swf Arithmetische Funktion (DE-588)4368429-4 s Natürliche Zahl (DE-588)4041357-3 s Darstellung (DE-588)4200624-7 s 1\p DE-604 Zahlentheorie (DE-588)4067277-3 s 2\p DE-604 Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 272 (DE-604)BV049758308 272 https://doi.org/10.1007/978-1-4613-8548-6 Verlag Volltext 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 |
| spellingShingle | Elliott, P. D. T. A. Arithmetic Functions and Integer Products Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics Mathematics Number theory Number Theory Mathematik Arithmetische Funktion (DE-588)4368429-4 gnd Zahlentheorie (DE-588)4067277-3 gnd Darstellung (DE-588)4200624-7 gnd Natürliche Zahl (DE-588)4041357-3 gnd |
| subject_GND | (DE-588)4368429-4 (DE-588)4067277-3 (DE-588)4200624-7 (DE-588)4041357-3 |
| title | Arithmetic Functions and Integer Products |
| title_auth | Arithmetic Functions and Integer Products |
| title_exact_search | Arithmetic Functions and Integer Products |
| title_full | Arithmetic Functions and Integer Products by P. D. T. A. Elliott |
| title_fullStr | Arithmetic Functions and Integer Products by P. D. T. A. Elliott |
| title_full_unstemmed | Arithmetic Functions and Integer Products by P. D. T. A. Elliott |
| title_short | Arithmetic Functions and Integer Products |
| title_sort | arithmetic functions and integer products |
| topic | Mathematics Number theory Number Theory Mathematik Arithmetische Funktion (DE-588)4368429-4 gnd Zahlentheorie (DE-588)4067277-3 gnd Darstellung (DE-588)4200624-7 gnd Natürliche Zahl (DE-588)4041357-3 gnd |
| topic_facet | Mathematics Number theory Number Theory Mathematik Arithmetische Funktion Zahlentheorie Darstellung Natürliche Zahl |
| url | https://doi.org/10.1007/978-1-4613-8548-6 |
| volume_link | (DE-604)BV049758308 |
| work_keys_str_mv | AT elliottpdta arithmeticfunctionsandintegerproducts |