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1998 | 85 | 2 | 179-195
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
85
Numer
2
Strony
179-195
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-10-10
Twórcy
  • Institut für Mathematik und angewandte Statistik, Universität für Bodenkultur, A-1180 Wien, Austria
autor
  • Institut für Mathematik und angewandte Statistik, Universität für Bodenkultur, A-1180 Wien, Austria
  • Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
autor
  • Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, Michigan 48109-1109, U.S.A.
Bibliografia
  • [1] N. Bourbaki, Algebra II, Springer, Berlin, 1990.
  • [2] K. Corrádi and I. Kátai, A comment on K. S. Gangadharan's paper 'Two classical lattice point problems', Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 17 (1967), 89-97 (in Hungarian).
  • [3] M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, Lecture Notes in Math. 1651, Springer, Berlin, 1997.
  • [4] J. L. Hafner, New omega theorems for two classical lattice point problems, Invent. Math. 63 (1981), 181-186.
  • [5] J. L. Hafner, On the average order of a class of arithmetical functions, J. Number Theory 15 (1982), 36-76.
  • [6] K. S. Gangadharan, Two classical lattice point problems, Proc. Cambridge Philos. Soc. 57 (1961), 699-721.
  • [7] G. H. Hardy, On the expression of a number as the sum of two squares, Quart. J. Math. 46 (1915), 263-283.
  • [8] G. H. Hardy, On Dirichlet's divisor problem, Proc. London Math. Soc. (2) 15 (1916), 1-25.
  • [9] D. R. Heath-Brown, The density of rational points on cubic surfaces, Acta Arith. 79 (1997), 17-30.
  • [10] E. Hlawka, J. Schoißengeier and R. Taschner, Geometric and Analytic Number Theory, Springer, Berlin, 1991.
  • [11] M. N. Huxley, Exponential sums and lattice points II, Proc. London Math. Soc. 66 (1993), 279-301.
  • [12] M. N. Huxley, Area, lattice points, and exponential sums, London. Math. Soc. Monographs (N.S.) 13, Oxford, 1996.
  • [13] A. E. Ingham, On two classical lattice point problems, Proc. Cambridge Philos. Soc. 36 (1940), 131-138.
  • [14] I. Kátai, The number of lattice points in a circle, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 8 (1965), 39-60.
  • [15] E. Krätzel, Lattice Points, Deutsch. Verlag Wiss., Berlin, 1988.
  • [16] G. Kuba, On sums of two k-th powers of numbers in residue classes II, Abh. Math. Sem. Univ. Hamburg 63 (1993), 87-95.
  • [17] W. G. Nowak, An Ω-estimate for the lattice rest of a convex planar domain, Proc. Roy. Soc. Edinburgh Sect. A 100 (1985), 295-299.
  • [18] W. G. Nowak, On the average order of the lattice rest of a convex planar domain, Proc. Cambridge Philos. Soc. 98 (1985), 1-4.
  • [19] W. G. Nowak, On sums of two k-th powers: a mean-square bound for the error term, Analysis 16 (1996), 297-304.
  • [20] W. G. Nowak, Sums of two k-th powers: an Omega estimate for the error term, Arch. Math. (Basel) 68 (1997), 27-35.
  • [21] J. D. Vaaler, Some extremal problems in Fourier analysis, Bull. Amer. Math. Soc. (2) 12 (1985), 183-216.
  • [22] J. G. van der Corput, Over roosterpunkten in het plate vlak, Thesis, Groningen, 1919.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-aav85i2p179bwm
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