Integral transforms of the Minkowski question mark function
The Minkowski question mark function F(x) arises as a real distribution function of rationals in the Farey (alias, Stern-Brocot or Calkin-Wilf) tree. In this thesis we introduce its three natural integral transforms: the dyadic period function G(z), defined in the cut plane; the dyadic zeta function...
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
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2008
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| Online Access: | https://eprints.nottingham.ac.uk/10641/ |
| _version_ | 1848791107590160384 |
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| author | Alkauskas, Giedrius |
| author_facet | Alkauskas, Giedrius |
| author_sort | Alkauskas, Giedrius |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | The Minkowski question mark function F(x) arises as a real distribution function of rationals in the Farey (alias, Stern-Brocot or Calkin-Wilf) tree. In this thesis we introduce its three natural integral transforms: the dyadic period function G(z), defined in the cut plane; the dyadic zeta function zeta_M(s), which is an entire function; the characteristic function m(t), which is an entire function as well. Each of them is a unique object, and is characterized by regularity properties and a functional equation, which reformulates in its own terms the functional equation for F(x). We study the interrelations among these three objects and F(x). It appears that the theory is completely parallel to the one for Maass wave forms for PSL_2(Z). One of the main purposes of this thesis is to clarify the nature of moments of the Minkowski question mark function. |
| first_indexed | 2025-11-14T18:23:15Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-10641 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T18:23:15Z |
| publishDate | 2008 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-106412025-02-28T11:09:02Z https://eprints.nottingham.ac.uk/10641/ Integral transforms of the Minkowski question mark function Alkauskas, Giedrius The Minkowski question mark function F(x) arises as a real distribution function of rationals in the Farey (alias, Stern-Brocot or Calkin-Wilf) tree. In this thesis we introduce its three natural integral transforms: the dyadic period function G(z), defined in the cut plane; the dyadic zeta function zeta_M(s), which is an entire function; the characteristic function m(t), which is an entire function as well. Each of them is a unique object, and is characterized by regularity properties and a functional equation, which reformulates in its own terms the functional equation for F(x). We study the interrelations among these three objects and F(x). It appears that the theory is completely parallel to the one for Maass wave forms for PSL_2(Z). One of the main purposes of this thesis is to clarify the nature of moments of the Minkowski question mark function. 2008 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en arr https://eprints.nottingham.ac.uk/10641/1/alkauskas_thesis.pdf Alkauskas, Giedrius (2008) Integral transforms of the Minkowski question mark function. PhD thesis, University of Nottingham. Minkowski question mark function period functions Farey tree moments of the distribution continued fractions |
| spellingShingle | Minkowski question mark function period functions Farey tree moments of the distribution continued fractions Alkauskas, Giedrius Integral transforms of the Minkowski question mark function |
| title | Integral transforms of the Minkowski question mark function |
| title_full | Integral transforms of the Minkowski question mark function |
| title_fullStr | Integral transforms of the Minkowski question mark function |
| title_full_unstemmed | Integral transforms of the Minkowski question mark function |
| title_short | Integral transforms of the Minkowski question mark function |
| title_sort | integral transforms of the minkowski question mark function |
| topic | Minkowski question mark function period functions Farey tree moments of the distribution continued fractions |
| url | https://eprints.nottingham.ac.uk/10641/ |