Involve: A Journal of Mathematics

  • Involve
  • Volume 2, Number 2 (2009), 121-159.

Generating and zeta functions, structure, spectral and analytic properties of the moments of the Minkowski question mark function

Giedrius Alkauskas

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Abstract

In this paper we are interested in moments of the Minkowski question mark function ?(x). It appears that, to some extent, the results are analogous to results obtained for objects associated with Maass wave forms: period functions, L-series, distributions. These objects can be naturally defined for ?(x) as well. Various previous investigations of ?(x) are mainly motivated from the perspective of metric number theory, Hausdorff dimension, singularity and generalizations. In this work it is shown that analytic and spectral properties of various integral transforms of ?(x) do reveal significant information about the question mark function. We prove asymptotic and structural results about the moments, calculate certain integrals which involve ?(x), define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects.

Article information

Source
Involve, Volume 2, Number 2 (2009), 121-159.

Dates
Received: 29 January 2008
Accepted: 29 December 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513799137

Digital Object Identifier
doi:10.2140/involve.2009.2.121

Mathematical Reviews number (MathSciNet)
MR2501334

Zentralblatt MATH identifier
1170.11028

Subjects
Primary: 11A55: Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 26A30: Singular functions, Cantor functions, functions with other special properties
Secondary: 11F99: None of the above, but in this section

Keywords
Minkowski question mark function Farey tree period functions distribution moments

Citation

Alkauskas, Giedrius. Generating and zeta functions, structure, spectral and analytic properties of the moments of the Minkowski question mark function. Involve 2 (2009), no. 2, 121--159. doi:10.2140/involve.2009.2.121. https://projecteuclid.org/euclid.involve/1513799137


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