Details

Autor(en) / Beteiligte

• Yuan, Wang

Titel

Diophantine Equations and Inequalities in Algebraic Number Fields [Elektronische Ressource]

Ort / Verlag

Berlin, Heidelberg : Springer Berlin Heidelberg

Erscheinungsjahr

1991

Link zum Volltext

• Volltext

Link zu anderen Inhalten

• Volltext

Beschreibungen/Notizen

• The circle method has its genesis in a paper of Hardy and Ramanujan (see [Hardy 1])in 1918concernedwiththepartitionfunction andtheproblemofrepresenting numbers as sums ofsquares. Later, in a series of papers beginning in 1920entitled "some problems of'partitio numerorum''', Hardy and Littlewood (see [Hardy 1]) created and developed systematically a new analytic method, the circle method in additive number theory. The most famous problems in additive number theory, namely Waring's problem and Goldbach's problem, are treated in their papers. The circle method is also called the Hardy-Littlewood method. Waring's problem may be described as follows: For every integer k 2 2, there is a number s= s( k) such that every positive integer N is representable as (1) where Xi arenon-negative integers. This assertion wasfirst proved by Hilbert [1] in 1909. Using their powerful circle method, Hardy and Littlewood obtained a deeper result on Waring's problem. They established an asymptotic formula for rs(N), the number of representations of N in the form (1), namely k 1 provided that 8 2 (k - 2)2 - +5. Here