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Generating and
zeta functions, structure, spectral and analytic properties
of the moments of the Minkowski question mark function
Giedrius Alkauskas
Vol. 2 (2009), No. 2, 121–159
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Abstract |
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In this paper we are interested in moments of the Minkowski question mark function . It appears that, to some extent, the results are analogous to results obtained for objects associated with Maass wave forms: period functions, -series, distributions. These objects can be naturally defined for as well. Various previous investigations of 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 do reveal significant information about the question mark function. We prove asymptotic and structural results about the moments, calculate certain integrals which involve , define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects. |
Keywords
Minkowski question mark function, Farey tree, period
functions, distribution moments
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Mathematical Subject Classification 2000
Primary: 11A55, 11M41, 26A30
Secondary: 11F99
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Milestones
Received: 29 January 2008
Accepted: 29 December 2008
Published: 7 May 2009
Communicated by Ken Ono
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Authors |
| Giedrius Alkauskas | |
| Vilnius University The Department of Mathematics and Informatics Naugarduko 24 Vilnius Lithuania |
The School of Mathematical
Sciences The University of Nottingham University Park, Nottingham NG7 2RD United Kingdom |
| http://alkauskas.ten.lt | |
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