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Fractional part sums and lattice points

Published online by Cambridge University Press:  20 January 2009

Werner Georg Nowak
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Abstract

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The objective of this article are sums S(M)=∑n;ψ(Mf(n/M)) where ψdenotes essentially the fractional part minus ½, f is a C4-function with fn nonvanishing, and summation is extended over an interval of order M. For S(M) an Ω-estimate and a mean-square bound is obtained. Applications to problems concerning the number of lattice points in large planar domains are discussed.

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 41 , Issue 3 , October 1998 , pp. 497 - 515
Copyright
Copyright © Edinburgh Mathematical Society 1998

Footnotes

*

This article is part of a research project supported by the Austrian Science Foundation (Nr. P 9892-PHY).

References

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. Branton, M. and Sargos, P., Points entiers au voisinage d'une courbe plane à très faible courbure, Bull. Sci. Math. 118 (1994), 15–28.Google Scholar
3. Chandrasekharan, K. and Narasimhan, R., On the mean value of the error term for a class of arithmetic functions, Acta Math. 112 (1964), 41–67.10.1007/BF02391764CrossRefGoogle Scholar
4. Van der Corput, J. G., Zahlentheoretische Abschätzungen mit Anwendung auf Gitterpunktprobleme, Math. Z. 17 (1923), 250–259.CrossRefGoogle Scholar
5. Van der Corput, J. G., Neue zahlentheoretische Abschätzungen, Math. Ann. 89 (1923), 215–254.CrossRefGoogle Scholar
6. Graham, S. W. and Kolesnik, G., Van der Corput's method of exponential sums (Cambridge University Press, Cambridge, 1991).10.1017/CBO9780511661976CrossRefGoogle Scholar
7. Hafner, J. L., New omega theorems for two classical lattice point problems, Invent. Math. 63(1981), 181–186.CrossRefGoogle Scholar
8. Hafner, J. L., On the average order of a class of arithmetical functions, J. Number Theory 15 (1982), 36–76.CrossRefGoogle Scholar
9. Hlawka, E., Über die Zetafunktion konvexer Körper, Monatsh Math. 54 (1950), 100–107.CrossRefGoogle Scholar
10. Huxley, M. N., Exponential sums and rounding error, J. London Math. Soc. (2) 43 (1991), 367–384.CrossRefGoogle Scholar
11. Huxley, M. N., Exponential sums and lattice points II, Proc. London Math. Soc. 66 (1993), 279–301.CrossRefGoogle Scholar
12. Huxley, M. N., The mean lattice point discrepancy, Proc. Edinburgh Math. Soc. 38 (1995), 523–531.10.1017/S0013091500019313CrossRefGoogle Scholar
13. Huxley, M. N., Area, lattice points, and exponential sums (LMS Monographs, New Ser. 13, Oxford 1996).CrossRefGoogle Scholar
14. Huxley, M. N. and Sargos, P., Points entiers au voisinage d'une courbe plane de classe C n, Acta Arithm. 69 (1995), 359–366.10.4064/aa-69-4-359-366CrossRefGoogle Scholar
15. Iwaniec, H. and Mozzochi, C. J., On the divisor and circle problems, J. Number Theory 29 (1988), 60–93.10.1016/0022-314X(88)90093-5CrossRefGoogle Scholar
16. Kátai, I., The number of lattice points in a circle, Ann. Univ. Sci. Budap. Rolando Eötvos, Sect. Math. 8 (1965), 39–60.Google Scholar
17. Krätzel, E., Lattice Points (Dt. Verl. d. Wiss., Berlin, 1988).Google Scholar
18. Kuipers, L. and Niederreiter, H., Uniform distribution of sequences (J. Wiley & Sons, New York 1974).Google Scholar
19. Müller, W., On the average order of the lattice rest of a convex body, Acta Arithm. 80 (1997), 89–100.CrossRefGoogle Scholar
20. Nowak, W. G., Einige Beiträge zur Theorie der Gitterpunkte, in Zahlentheoretische Analysis (Seminar 1980–1982, ed. Hlawka, E., Springer LNM 1114, 1985), pp. 98–117.CrossRefGoogle Scholar
21. Nowak, W. G., An Ω-estimate for the lattice rest of a convex planar domain, Proc. Roy. Soc. Edinburgh Sect. A 100 (1985), 295–299.CrossRefGoogle Scholar
22. Nowak, W. G., On the average order of the lattice rest of a convex planar domain, Proc. Cambridge Philos. Soc. 98 (1985), 1–4.CrossRefGoogle Scholar
23. Redmond, D., Mean value theorems for a class of Dirichlet series, Pacific J. Math. 78 (1978), 191–231.CrossRefGoogle Scholar
24. Swinnerton-Dyer, H. P. F., The number of lattice points on a convex curve, J Number Theory 6 (1974), 128–135.CrossRefGoogle Scholar
25. Vaaler, J. D., Some extremal problems in Fourier analysis, Bull. Amer. Math. Soc. (2) 12 (1985), 183–216.10.1090/S0273-0979-1985-15349-2CrossRefGoogle Scholar
26. Vinogradov, I. M., Selected works (eds. Faddeev, L. D. et al. , Springer, Berlin-Heidelberg 1985).10.1007/978-3-642-15086-9CrossRefGoogle Scholar