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

Abstract

In this paper we are interested in moments of the Minkowski question mark function $?\left(x\right)$. It appears that, to some extent, the results are analogous to results obtained for objects associated with Maass wave forms: period functions, $L$-series, distributions. These objects can be naturally defined for $?\left(x\right)$ as well. Various previous investigations of $?\left(x\right)$ 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 $?\left(x\right)$ do reveal significant information about the question mark function. We prove asymptotic and structural results about the moments, calculate certain integrals which involve $?\left(x\right)$, define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects.