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On the Distribution of Integer Ideals in Algebraic Number Fields

✍ Scribed by Werner Georg

Publisher: John Wiley and Sons
Year: 1993
Tongue: English
Weight: 585 KB
Volume: 161
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.19931610107

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✦ Synopsis

fields, the problem is essentially a planar lattice point problem (cf. ZAGIER [17]). To this, the deep results of HUXLEY [3], [4] can be applied to get

For cubic fields, W. MULLER [12] proved that

43

(h the class number), using a deep exponential sum technique due to KOLESNIK [7].

every n 2 3, and for any K , independently of its algebraic properties.

The aim of the present article is to improve LANDAU'S classical upper bound (1.2) for

Theorem. For any algebraic number field K of degree [K

, and any ideal class U,

2 8 10 q x l --+n n(5n+2)(log x)5n+2) for 3 5 n < 6 , 2 3 2

O(x'-"+2"2(logx)q for n 2 7 .

A(x, U) = Ax + (Throughout the paper, all constants implied in the symbols 0 , <<, and x depend only on K and %' .

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