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An asymptotic formula for the moments of the Minkowski question mark function in the interval [0, 1]

Lithuanian Mathematical Journal volume 48, pages 357–367 (2008)Cite this article

Abstract

In this paper, we prove an asymptotic formula for the moments of the Minkowski question mark function, which describes the distribution of rationals in the Farey tree. The main idea is to demonstrate that a certain variation of a Laplace method is applicable in this problem, and hence the task reduces to a number of technical calculations.

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References

  1. An Exhaustive Bibliography on the Minkowski Question Mark Function, available from Internet: http://www.maths.nottingham.ac.uk/personal/pmxga2/minkowski.htm.

  2. G. Alkauskas, Generating and zeta functions, structure, spectral and analytic properties of the moments of Minkowski question mark function (submitted), arXiv:0801.0056.

  3. G. Alkauskas, Minkowski question mark function and its generalizations, associated with p-continued fractions: fractals, explicit series for the dyadic period function and moments (submitted), arXiv:0805.1717.

  4. G. Alkauskas, The moments of Minkowski question mark function: the dyadic period function (submitted), arXiv:0801.0051.

  5. N. Calkin and H. Wilf, Recounting the rationals, Am. Math. Mon., 107:360–363, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Denjoy, Sur une fonction réelle de Minkowski, J. Math. Pures Appl., 17:105–151, 1938.

    MATH  Google Scholar 

  7. P.J. Grabner, P. Kirschenhofer, and R.F. Tichy, Combinatorial and arithmetical properties of linear numeration systems, Combinatorica, 22(2):245–267, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Ya. Khinchin, Continued Fractions, The University of Chicago Press, 1964.

  9. M.A. Lavrent'ev and B. V. Shabat, Methods of the Theory of Functions of a Complex Variable, Nauka, Moscow, 1987 (in Russian).

    MATH  Google Scholar 

  10. J.B. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms, Invent. Math., 127(2):271–306, 1997.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. The School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK

    G. Alkauskas

  2. The Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius, Lithuania

    G. Alkauskas

Authors
  1. G. Alkauskas
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Correspondence to G. Alkauskas.

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Dedicated to Antanas Laurinčikas on the occasion of his 60th birthday

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Alkauskas, G. An asymptotic formula for the moments of the Minkowski question mark function in the interval [0, 1]. Lith Math J 48, 357–367 (2008). https://doi.org/10.1007/s10986-008-9027-3

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  • DOI: https://doi.org/10.1007/s10986-008-9027-3

Keywords

  • Minkowski question mark function
  • Laplace method of asymptotic expansion
  • Farey tree
  • moments of distribution