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

• Autorzy: M. Kühleitner , W. G. Nowak , J. Schoissengeier , T. D. Wooley
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11N37: Asymptotic results on arithmetic functions
• 11P21: Lattice points in specified regions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1998
• Tom: 85
• Numer: 2
• Strony: 179-195

Daty

wydano

1998

otrzymano

1997-10-10

Twórcy

autor

M. Kühleitner

• Institut für Mathematik und angewandte Statistik, Universität für Bodenkultur, A-1180 Wien, Austria

autor

W. G. Nowak

• Institut für Mathematik und angewandte Statistik, Universität für Bodenkultur, A-1180 Wien, Austria

autor

J. Schoissengeier

• Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria

autor

T. D. Wooley

• Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, Michigan 48109-1109, U.S.A.

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