Probabilistic Number Theory I: Mean-Value Theorems - Hardcover

Hardcover

ISBN 10: 3540904379 ISBN 13: 9783540904373

Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K, 1979

Synopsis

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.

Edition details

• Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
• Publication date: 1979
• Language: English
• ISBN 10: 3540904379
• ISBN 13: 9783540904373
• Binding: Hardcover
• Number of pages: 393