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

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

Acta Arithmetica (1998)

  • Volume: 85, Issue: 2, page 179-195
  • ISSN: 0065-1036

Abstract

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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 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 ( l o g l o g 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.

How to cite

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M. Kühleitner, et al. "On sums of two cubes: an Ω₊-estimate for the error term." Acta Arithmetica 85.2 (1998): 179-195. <http://eudml.org/doc/207161>.

@article{M1998,
abstract = {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.},
author = {M. Kühleitner, W. G. Nowak, J. Schoissengeier, T. D. Wooley},
journal = {Acta Arithmetica},
keywords = {sums of two cubes; estimate for the error term; asymptotical formula; summatory function},
language = {eng},
number = {2},
pages = {179-195},
title = {On sums of two cubes: an Ω₊-estimate for the error term},
url = {http://eudml.org/doc/207161},
volume = {85},
year = {1998},
}

TY - JOUR
AU - M. Kühleitner
AU - W. G. Nowak
AU - J. Schoissengeier
AU - T. D. Wooley
TI - On sums of two cubes: an Ω₊-estimate for the error term
JO - Acta Arithmetica
PY - 1998
VL - 85
IS - 2
SP - 179
EP - 195
AB - 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.
LA - eng
KW - sums of two cubes; estimate for the error term; asymptotical formula; summatory function
UR - http://eudml.org/doc/207161
ER -

References

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