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1998 | 85 | 2 | 179-195
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On sums of two cubes: an Ω₊-estimate for the error term

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The arithmetic function $r_k(n)$ 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 $r_k(n)$ leads in a natural way to a certain error term $P_{𝓓_k}(t)$ which is known to be $O(t^{1/4})$ in mean-square. In this article it is proved that $P_{𝓓₃}(t) = Ω₊(t^{1/4}(loglog t)^{1/4})$ 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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  • Institut für Mathematik und angewandte Statistik, Universität für Bodenkultur, A-1180 Wien, Austria
  • Institut für Mathematik und angewandte Statistik, Universität für Bodenkultur, A-1180 Wien, Austria
  • Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
  • Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, Michigan 48109-1109, U.S.A.
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