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

Abstract

In this paper we are interested in moments of Minkowski question mark function ?(x). It appears that, to certain extent, the results are analogous to the results obtained for objects associated with Maass wave forms: period functions, L-series, distributions, spectral properties. These objects can be naturally defined for ?(x) as well. Despite the fact that there are various nice results about the nature of ?(x), these investigations 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 involving ?(x), define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects. At the end of the paper it is shown that certain object associated with ?(x) establish a bridge between realms of imaginary and real quadratic irrationals.

34 pages, 4 figures (submitted 01/2008). Minor revisions and typos. A graph of dyadic zeta function on the critical line was added. Theorem 3 was strengthened

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Cited by in corpus (8)

• The Minkowski question mark function: explicit series for the dyadic period function and moments
• The moments of Minkowski question mark function: the dyadic period function
• Asymptotic formula for the moments of Minkowski question mark function in the interval [0,1]
• Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas
• On the Minkowski Measure
• Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function
• Fourier-Stieltjes coefficients of the Minkowski question mark function
• The Minkowski $?(x)$ function, a class of singular measures, quasi-modular and mean-modular forms. I