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THE MOMENTS OF MINKOWSKI QUESTION MARK FUNCTION: THE DYADIC PERIOD FUNCTION

Published online by Cambridge University Press:  30 July 2009

GIEDRIUS ALKAUSKAS
GIEDRIUS ALKAUSKAS*
Affiliation:
The Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius, Lithuania and School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK e-mail: giedrius.alkauskas@gmail.com
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Abstract

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The Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert–Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern–Brocot tree. Surprisingly, the Eisenstein series G2(z) does manifest in both real and p-adic cases.

Keywords

Primary – 11A5526A3011F03Secondary – 33C10

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 1 , January 2010, pp. 41 - 64
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

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