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Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function

The Ramanujan Journal volume 25, pages 359–367 (2011)Cite this article

Abstract

This paper continues investigations on various integral transforms of the Minkowski question mark function. In this work we finally establish the long-sought formula for the moments, which does not explicitly involve regular continued fractions, though it has a hidden nice interpretation in terms of semi-regular continued fractions. The proof is self-contained and does not rely on previous results by the author.

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Authors and Affiliations

  1. Institute of Mathematics, Department of Integrative Biology, Universität für Bodenkultur Wien, Gregor Mendel-Straße 33, 1180, Vienna, Austria

    Giedrius Alkauskas

  2. Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania

    Giedrius Alkauskas

Authors
  1. Giedrius Alkauskas
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Corresponding author

Correspondence to Giedrius Alkauskas.

Additional information

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

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Alkauskas, G. Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function. Ramanujan J 25, 359–367 (2011). https://doi.org/10.1007/s11139-011-9307-0

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  • DOI: https://doi.org/10.1007/s11139-011-9307-0

Keywords

  • The Minkowski question mark function
  • Moments of distribution
  • Periods
  • Bessel functions
  • Semi-regular continued fractions
  • The Farey tree

Mathematics Subject Classification (2000)

  • 11A55
  • 26A30
  • 33C10
  • 11F67