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A special case of Vinogradov's mean value theorem

100%

Vaughan R., Wooley T.

Acta Arithmetica

1997

tom 79

nr 3

193-204

Additive representation in thin sequences, III: asymptotic formulae

81%

Brüdern J., Kawada K., Wooley T.

Acta Arithmetica

2001

tom 100

nr 3

267-289

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

81%

Kühleitner M., Nowak W., Schoissengeier J., Wooley T.

Acta Arithmetica

1998

tom 85

nr 2

179-195

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.

Ograniczanie wyników

3 Acta Arithmetica

3 Wooley T.

1 Brüdern J.

1 Kawada K.

1 Kühleitner M.

1 Nowak W.

1 Schoissengeier J.

1 Vaughan R.

1 2001

1 1998

1 1997

Wyniki wyszukiwania

A special case of Vinogradov's mean value theorem

Additive representation in thin sequences, III: asymptotic formulae

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