Giedrius Alkauskas

THE MOMENTS OF MINKOWSKI QUESTION MARK FUNCTION: THE DYADIC PERIOD FUNCTION

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 \ (1, ∞). The exponential generating function satisfies an 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 2 ( z ) does manifest in both real and p -adic cases.

Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function

This paper continues investigations on various integral transforms of the Minkowski question mark function. In this work we finally establish the long-sought formula for the moments, which does not explicitly involve regular continued fractions, though it has a hidden nice interpretation in terms of semi-regular continued fractions. The proof is self-contained and does not rely on previous results by the author.

Multi-variable translation equation which arises from homothety

In many regular cases, there exists a (properly defined) limit of iterations of a function in several real variables, and this limit satisfies the functional equation

Algebraic and abelian solutions to the projective translation equation

{(1-z)\phi({\bf x})=\phi(\phi({\bf x}z)(1-z)/z)}

An asymptotic formula for the moments of the Minkowski question mark function in the interval [0, 1]

; here z is a scalar and x is a vector. This is a special case of a well-known translation equation. In this paper we present a complete solution to this functional equation when

The modular group and words in its two generators

{\phi}