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ArticleOriginal scientific text

Title

On sums of two cubes: an Ω₊-estimate for the error term

Authors 1, 1, 2, 3

Affiliations

  1. Institut für Mathematik und angewandte Statistik, Universität für Bodenkultur, A-1180 Wien, Austria
  2. Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
  3. Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, Michigan 48109-1109, U.S.A.

Abstract

The arithmetic function rk(n) counts the number of ways to write a natural number n as a sum of two kth powers (k ≥ 2 fixed). The investigation of the asymptotic behaviour of the Dirichlet summatory function of rk(n) leads in a natural way to a certain error term P_k(t) which is known to be O(t14) in mean-square. In this article it is proved that P₃(t)=Ω₊(t14(loglogt)14) as t → ∞. Furthermore, it is shown that a similar result would be true for every fixed k > 3 provided that a certain set of algebraic numbers contains a sufficiently large subset which is linearly independent over ℚ.

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Acta Arithmetica
1998/85/2
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Pages:
179-195
Main language of publication
English
Received
1997-10-10
Published
1998
Exact and natural sciences