Probabilistic Number Theory II: Central Limit Theorems
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer US
1980
|
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
240 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | In this volume we study the value distribution of arithmetic functions, allowing unbounded renormalisations. The methods involve a synthesis of Probability and Number Theory; sums of independent infinitesimal random variables playing an important role. A central problem is to decide when an additive arithmetic function fin) admits a renormalisation by real functions a(x) and {3(x) > 0 so that asx ~ 00 the frequencies vx(n;f (n) - a(x) :s;; z {3 (x) ) converge weakly; (see Notation). In contrast to volume one we allow {3(x) to become unbounded with x. In particular, we investigate to what extent one can simulate the behaviour of additive arithmetic functions by that of sums of suitably defined independent random variables. This fruiful point of view was introduced in a 1939 paper of Erdos and Kac. We obtain their (now classical) result in Chapter 12. Subsequent methods involve both Fourier analysis on the line, and the application of Dirichlet series. Many additional topics are considered. We mention only: a problem of Hardy and Ramanujan; local properties of additive arithmetic functions; the rate of convergence of certain arithmetic frequencies to the normal law; the arithmetic simulation of all stable laws. As in Volume I the historical background of various results is discussed, forming an integral part of the text. In Chapters 12 and 19 these considerations are quite extensive, and an author often speaks for himself |
| Beschreibung: | 1 Online-Ressource (375p) |
| ISBN: | 9781461299929 9781461299943 |
| DOI: | 10.1007/978-1-4612-9992-9 |
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| 500 | |a In this volume we study the value distribution of arithmetic functions, allowing unbounded renormalisations. The methods involve a synthesis of Probability and Number Theory; sums of independent infinitesimal random variables playing an important role. A central problem is to decide when an additive arithmetic function fin) admits a renormalisation by real functions a(x) and {3(x) > 0 so that asx ~ 00 the frequencies vx(n;f (n) - a(x) :s;; z {3 (x) ) converge weakly; (see Notation). In contrast to volume one we allow {3(x) to become unbounded with x. In particular, we investigate to what extent one can simulate the behaviour of additive arithmetic functions by that of sums of suitably defined independent random variables. This fruiful point of view was introduced in a 1939 paper of Erdos and Kac. We obtain their (now classical) result in Chapter 12. Subsequent methods involve both Fourier analysis on the line, and the application of Dirichlet series. Many additional topics are considered. We mention only: a problem of Hardy and Ramanujan; local properties of additive arithmetic functions; the rate of convergence of certain arithmetic frequencies to the normal law; the arithmetic simulation of all stable laws. As in Volume I the historical background of various results is discussed, forming an integral part of the text. In Chapters 12 and 19 these considerations are quite extensive, and an author often speaks for himself | ||
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| series2 | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics |
| spelling | Elliott, P. D. T. A. Verfasser aut Probabilistic Number Theory II Central Limit Theorems by P. D. T. A. Elliott New York, NY Springer US 1980 1 Online-Ressource (375p) txt rdacontent c rdamedia cr rdacarrier Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 240 In this volume we study the value distribution of arithmetic functions, allowing unbounded renormalisations. The methods involve a synthesis of Probability and Number Theory; sums of independent infinitesimal random variables playing an important role. A central problem is to decide when an additive arithmetic function fin) admits a renormalisation by real functions a(x) and {3(x) > 0 so that asx ~ 00 the frequencies vx(n;f (n) - a(x) :s;; z {3 (x) ) converge weakly; (see Notation). In contrast to volume one we allow {3(x) to become unbounded with x. In particular, we investigate to what extent one can simulate the behaviour of additive arithmetic functions by that of sums of suitably defined independent random variables. This fruiful point of view was introduced in a 1939 paper of Erdos and Kac. We obtain their (now classical) result in Chapter 12. Subsequent methods involve both Fourier analysis on the line, and the application of Dirichlet series. Many additional topics are considered. We mention only: a problem of Hardy and Ramanujan; local properties of additive arithmetic functions; the rate of convergence of certain arithmetic frequencies to the normal law; the arithmetic simulation of all stable laws. As in Volume I the historical background of various results is discussed, forming an integral part of the text. In Chapters 12 and 19 these considerations are quite extensive, and an author often speaks for himself Mathematics Number theory Number Theory Mathematik Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 240 (DE-604)BV049758308 240 https://doi.org/10.1007/978-1-4612-9992-9 Verlag Volltext |
| spellingShingle | Elliott, P. D. T. A. Probabilistic Number Theory II Central Limit Theorems Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics Mathematics Number theory Number Theory Mathematik |
| title | Probabilistic Number Theory II Central Limit Theorems |
| title_auth | Probabilistic Number Theory II Central Limit Theorems |
| title_exact_search | Probabilistic Number Theory II Central Limit Theorems |
| title_full | Probabilistic Number Theory II Central Limit Theorems by P. D. T. A. Elliott |
| title_fullStr | Probabilistic Number Theory II Central Limit Theorems by P. D. T. A. Elliott |
| title_full_unstemmed | Probabilistic Number Theory II Central Limit Theorems by P. D. T. A. Elliott |
| title_short | Probabilistic Number Theory II |
| title_sort | probabilistic number theory ii central limit theorems |
| title_sub | Central Limit Theorems |
| topic | Mathematics Number theory Number Theory Mathematik |
| topic_facet | Mathematics Number theory Number Theory Mathematik |
| url | https://doi.org/10.1007/978-1-4612-9992-9 |
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