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The asymptotic behaviour of the mean-square of fractional part sums

Published online by Cambridge University Press:  20 January 2009

Manfred Kühleitner  and
Werner Georg Nowak
Manfred Kühleitner
Affiliation:
Institut für Mathematik u. Ang. Stat., Universität für Bodenkultur, A-1180 Wien, Austria (kleitner@mail.boku.ac.at; nowak@mail.boku.ac.at)
Werner Georg Nowak
Affiliation:
Institut für Mathematik u. Ang. Stat., Universität für Bodenkultur, A-1180 Wien, Austria (kleitner@mail.boku.ac.at; nowak@mail.boku.ac.at)
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Abstract

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In this article we consider sums S(t) = Σnψ (tf(n/t)), where ψ denotes, essentially, the fractional part minus ½ f is a C4-function with f″ non-vanishing, and summation is extended over an interval of order t. For the mean-square of S(t), an asymptotic formula is established. If f is algebraic this can be sharpened by the indication of an error term.

Keywords

fractional partsmean squarelattice pointsexponential sums
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 2 , June 2000 , pp. 309 - 323
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1. Bombieri, E. and Iwaniec, H., On the order of ζ(½ + it), Ann. Scuola Norm. Sup. Pisa, Ser. IV 13 (1986), 449–472.Google Scholar
2. Cramér, H., Über zwei Sätze von Herrn G. H. Hardy, Math. Z. 15 (1922), 200–210.CrossRefGoogle Scholar
3. Graham, S. W. and Kolesnik, G., Van der Corput's method of exponential sums (Cambridge University Press, Cambridge, 1991).Google Scholar
4. Huxley, M. N., Exponential sums and rounding error, J. Lond. Math. Soc. 43 (1991), 367–384.CrossRefGoogle Scholar
5. Huxley, M. N., Exponential sums and lattice points II, Proc. Lond. Math. Soc. 66 (1993), 279–301.CrossRefGoogle Scholar
6. Huxley, M. N., Area, lattice points, and exponential sums, LMS Monographs, New Series, vol. 13 (Oxford, 1996).CrossRefGoogle Scholar
7. Iwaniec, H. and Mozzochi, C. J., On the divisor and circle problems, J. Number Theory 29 (1988), 60–93.CrossRefGoogle Scholar
8. Kátai, I., The number of lattice points in a circle, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 8 (1965), 39–60.Google Scholar
9. Krätzel, E., Lattice points (Kluwer, Dordrecht, 1988).Google Scholar
10. Kühleitner, M., On sums of two kth powers: an asymptotic formula for the mean-square of the error term, Acta Arithm. 92 (2000), 263–276.CrossRefGoogle Scholar
11. Kühleitner, M., On differences of two kth powers: an asymptotic formula for the mean-square of the error term, J. Number Theory 76 (1999), 22–44.CrossRefGoogle Scholar
12. Kuipers, L. and Niederreiter, H., Uniform distribution of sequences (Wiley, New York, 1974).Google Scholar
13. Landau, E., Über die Gitterpunkte in einem Kreise (Vierte Mitteilung), Nachr. Königl. Ges. Wiss. Göttingen (Math.-Phys. Kl.) 1923 (1923), 58–65.Google Scholar
14. Landau, E., Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale (Chelsea, New York, 1949).Google Scholar
15. Nowak, W. G., Fractional part sums and lattice points, Proc. Edinb. Math. Soc. 41 (1998), 497–515.CrossRefGoogle Scholar
16. Swinnerton-Dyer, H. P. F., The number of lattice points on a convex curve, J. Number Theory 6 (1974), 128–135.CrossRefGoogle Scholar
17. Vaaler, J. D., Some extremal problems in Fourier analysis, Bull. Am. Math. Soc. 12 (1985), 183–216.CrossRefGoogle Scholar
18. Van Der Corput, J. G., Zahlentheoretische Abschätzungen mit Anwendung auf Gitterpunktprobleme, Math. Z. 17 (1923), 250–259.CrossRefGoogle Scholar
19. Van Der Corput, J. G., Neue zahlentheoretische Abschätzungen, Math. Ann. 89 (1923), 215–254.CrossRefGoogle Scholar
20. Vinogradov, I. M., Selected works (ed. Faddeev, L. D., Gamkrelidze, R. V., Karatsuba, A. A., Mardzhanishvili, K. K. and Mishchenko, E. F.) (Springer, Berlin, 1985).CrossRefGoogle Scholar