Probabilistic Number Theory I: Mean-Value Theorems
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer New York
1979
|
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
239 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | In 1791 Gauss made the following assertions (collected works, Vol. 10, p.ll, Teubner, Leipzig 1917): Primzahlen unter a ( = 00 ) a la Zahlen aus zwei Factoren lla· a la (warsch.) aus 3 Factoren 1 (lla)2a --- 2 la et sic in info In more modern notation, let 1tk(X) denote the number of integers not exceeding x which are made up of k distinct prime factors, k = 1, 2, .... Then his assertions amount to the asymptotic estimate x (log log X)k-l ( ) 1tk X '" --"';"'-"---"::--:-'-,- (x-..oo). log x (k-1)! The case k = 1, known as the Prime Number Theorem, was independently established by Hadamard and de la Vallee Poussin in 1896, just over a hundred years later. The general case was deduced by Landau in 1900; it needs only an integration by parts. Nevertheless, one can scarcely say that Probabilistic Number Theory began with Gauss. In 1914 the Indian original mathematician Srinivasa Ramanujan arrived in England. Six years of his short life remained to him during which he wrote, amongst other things, five papers and two notes jointly with G. H. Hardy |
| Beschreibung: | 1 Online-Ressource (393p) |
| ISBN: | 9781461299899 9781461299912 |
| ISSN: | 0072-7830 |
| DOI: | 10.1007/978-1-4612-9989-9 |
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| 500 | |a In 1791 Gauss made the following assertions (collected works, Vol. 10, p.ll, Teubner, Leipzig 1917): Primzahlen unter a ( = 00 ) a la Zahlen aus zwei Factoren lla· a la (warsch.) aus 3 Factoren 1 (lla)2a --- 2 la et sic in info In more modern notation, let 1tk(X) denote the number of integers not exceeding x which are made up of k distinct prime factors, k = 1, 2, .... Then his assertions amount to the asymptotic estimate x (log log X)k-l ( ) 1tk X '" --"';"'-"---"::--:-'-,- (x-..oo). log x (k-1)! The case k = 1, known as the Prime Number Theorem, was independently established by Hadamard and de la Vallee Poussin in 1896, just over a hundred years later. The general case was deduced by Landau in 1900; it needs only an integration by parts. Nevertheless, one can scarcely say that Probabilistic Number Theory began with Gauss. In 1914 the Indian original mathematician Srinivasa Ramanujan arrived in England. Six years of his short life remained to him during which he wrote, amongst other things, five papers and two notes jointly with G. H. Hardy | ||
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| indexdate | 2024-07-10T01:21:07Z |
| institution | BVB |
| isbn | 9781461299899 9781461299912 |
| issn | 0072-7830 |
| language | English |
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| spelling | Elliott, P. D. T. A. Verfasser aut Probabilistic Number Theory I Mean-Value Theorems by P. D. T. A. Elliott New York, NY Springer New York 1979 1 Online-Ressource (393p) txt rdacontent c rdamedia cr rdacarrier Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 239 0072-7830 In 1791 Gauss made the following assertions (collected works, Vol. 10, p.ll, Teubner, Leipzig 1917): Primzahlen unter a ( = 00 ) a la Zahlen aus zwei Factoren lla· a la (warsch.) aus 3 Factoren 1 (lla)2a --- 2 la et sic in info In more modern notation, let 1tk(X) denote the number of integers not exceeding x which are made up of k distinct prime factors, k = 1, 2, .... Then his assertions amount to the asymptotic estimate x (log log X)k-l ( ) 1tk X '" --"';"'-"---"::--:-'-,- (x-..oo). log x (k-1)! The case k = 1, known as the Prime Number Theorem, was independently established by Hadamard and de la Vallee Poussin in 1896, just over a hundred years later. The general case was deduced by Landau in 1900; it needs only an integration by parts. Nevertheless, one can scarcely say that Probabilistic Number Theory began with Gauss. In 1914 the Indian original mathematician Srinivasa Ramanujan arrived in England. Six years of his short life remained to him during which he wrote, amongst other things, five papers and two notes jointly with G. H. Hardy Mathematics Mathematics, general Mathematik https://doi.org/10.1007/978-1-4612-9989-9 Verlag Volltext |
| spellingShingle | Elliott, P. D. T. A. Probabilistic Number Theory I Mean-Value Theorems Mathematics Mathematics, general Mathematik |
| title | Probabilistic Number Theory I Mean-Value Theorems |
| title_auth | Probabilistic Number Theory I Mean-Value Theorems |
| title_exact_search | Probabilistic Number Theory I Mean-Value Theorems |
| title_full | Probabilistic Number Theory I Mean-Value Theorems by P. D. T. A. Elliott |
| title_fullStr | Probabilistic Number Theory I Mean-Value Theorems by P. D. T. A. Elliott |
| title_full_unstemmed | Probabilistic Number Theory I Mean-Value Theorems by P. D. T. A. Elliott |
| title_short | Probabilistic Number Theory I |
| title_sort | probabilistic number theory i mean value theorems |
| title_sub | Mean-Value Theorems |
| topic | Mathematics Mathematics, general Mathematik |
| topic_facet | Mathematics Mathematics, general Mathematik |
| url | https://doi.org/10.1007/978-1-4612-9989-9 |
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