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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An Exhaustive Bibliography on the Minkowski Question Mark Function, available from Internet: http://www.maths.nottingham.ac.uk/personal/pmxga2/minkowski.htm.
G. Alkauskas, Generating and zeta functions, structure, spectral and analytic properties of the moments of Minkowski question mark function (submitted), arXiv:0801.0056.
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.
G. Alkauskas, The moments of Minkowski question mark function: the dyadic period function (submitted), arXiv:0801.0051.
N. Calkin and H. Wilf, Recounting the rationals, Am. Math. Mon., 107:360–363, 2000.
A. Denjoy, Sur une fonction réelle de Minkowski, J. Math. Pures Appl., 17:105–151, 1938.
P.J. Grabner, P. Kirschenhofer, and R.F. Tichy, Combinatorial and arithmetical properties of linear numeration systems, Combinatorica, 22(2):245–267, 2002.
A. Ya. Khinchin, Continued Fractions, The University of Chicago Press, 1964.
M.A. Lavrent'ev and B. V. Shabat, Methods of the Theory of Functions of a Complex Variable, Nauka, Moscow, 1987 (in Russian).
J.B. Lewis, Spaces of holomorphic functions equivalent to the even Maass cusp forms, Invent. Math., 127(2):271–306, 1997.
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
- Minkowski question mark function
- Laplace method of asymptotic expansion
- Farey tree
- moments of distribution