The moments of Minkowski question mark function: the dyadic period function

Abstract

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 C(0,infinity). The exponential generating function satisfies the 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 G_1(z) does manifest in both real and p-adic cases.

26 pages, 1 figure (submitted). The current paper is an essential revision of the previous version (September 2006-May 2007). Some results from an article arXiv:0801.0054 were merged into a new version

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• Orthogonal polynomials for Minkowski's question mark function
• Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function
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• Mean Row Values in $(u,v)$-Calkin-Wilf Trees
• The Minkowski $?(x)$ function, a class of singular measures, quasi-modular and mean-modular forms. I
• The Stern Sequence and Moments of Minkowski's Question Mark Function