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The average size of Ramanujan sums over quadratic number fields

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Abstract

This article is concerned with Ramanujan sums \({c_{\mathcal{I}_1}(\mathcal{I}),}\) where \({\mathcal{I},\mathcal{I}_1}\) are integral ideals in an arbitrary quadratic number field \({\mathbb{Q}(\sqrt{d}).}\) In particular, the asymptotic behavior of sums of \({c_{\mathcal{I}_1}(\mathcal{I}),}\) over both \({\mathcal{I}}\) and \({c_{\mathcal{I}_1}(\mathcal{I}),}\) is investigated.

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References

  1. Chandrasekharan K., Narasimhan R.: Functional equations with multiple Gamma factors and the average order of arithmetical functions. Ann. Math. 76, 93–136 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chan T.H., Kumchev A.V.: On sums of Ramanujan sums. Acta arithm. 152, 1–10 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Grytczuk A.: On Ramanujan sums on arithmetical semigroups. Tsukuba J. Math. 16, 315–319 (1992)

    MathSciNet  MATH  Google Scholar 

  4. E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, 2nd ed., Chelsea Publ. Co., (New York, 1948).

  5. Huxley M.N.: Exponential sums and lattice points III. Proc. London Math. Soc. 87, 591–609 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Huxley M.N.: Exponential sums and the Riemann zeta-function V. Proc. London Math. Soc. 90, 1–41 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Knopfmacher, Abstract analytic number theory, North Holland Publ. Co., (Amsterdam-Oxford, 1975).

  8. E. Krätzel, Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, (Berlin, 1981).

  9. E. Krätzel, Lattice points, VEB Deutscher Verlag der Wissenschaften, (Berlin, 1988).

  10. Krätzel E., Nowak W.G.: Lattice points in large convex bodies. Monatsh. Math. 112, 61–72 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Müller W.: On the asymptotic behaviour of the ideal counting function in quadratic number fields. Monatsh. Math. 108, 301–323 (1989)

    MATH  Google Scholar 

  12. W.G. Nowak, Higher order derivative tests for exponential sums incorporating the Discrete Hardy-Littlewood method, Acta Math. Hungar. 134/1 (2012), 12–28.

    Google Scholar 

  13. W.G. Nowak, On Ramanujan sums over the Gaussian integers. Math. Slovaca, to appear.

  14. E.C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., revised by D.R. Heath-Brown, University Press, (Oxford, 1986).

  15. D.B. Zagier, Zetafunktionen und quadratische Körper, Springer, (Berlin, 1981).

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Correspondence to Werner Georg Nowak.

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The author gratefully acknowledges support from the Austrian Science Fund (FWF) under project Nr. P20847-N18.

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Nowak, W.G. The average size of Ramanujan sums over quadratic number fields. Arch. Math. 99, 433–442 (2012). https://doi.org/10.1007/s00013-012-0442-7

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