# 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

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topM. 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 -

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