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

TL;DR

This paper explores the moments of Minkowski question mark function, revealing their spectral, analytic, and structural properties, and establishing connections with zeta functions, Fourier series, and quadratic irrationals.

Contribution

It introduces new integral transforms, defines an associated zeta function, and uncovers relations linking the question mark function to spectral theory and quadratic irrationals.

Findings

Derived asymptotic and structural results for moments.

Calculated integrals involving ?(x).

Established relations between ?(x) and quadratic irrationals.

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.