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

M. Kühleitner ; W. G. Nowak ; J. Schoissengeier ; T. D. Wooley

Acta Arithmetica, Tome 84 (1998), p. 179-195 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $P_{_k}(t)$ which is known to be $O(t^{1/4})$ in mean-square. In this article it is proved that $P_{₃}(t) = Ω₊(t^{1/4}(loglog t)^{1/4})$ 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.

Détail

Publié le : 1998-01-01

Zbl 0916.11052

EUDML-ID : urn:eudml:doc:207161

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