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

TL;DR

This paper studies the moments of the Minkowski question mark function, revealing their connection to dyadic period functions, integral equations with Bessel kernels, and p-adic distributions, highlighting deep links between real and p-adic number theory.

Contribution

It introduces a dyadic analogue of period functions for the Minkowski question mark function and explores their integral equations and p-adic distributions, revealing new structural insights.

Findings

The generating function satisfies an integral equation with a Bessel kernel.

Dyadic eigenfunctions are derived from a Hilbert-Schmidt operator.

The Eisenstein series G_1(z) appears in both real and p-adic contexts.

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.