MATHEMATICS OF COMPUTATION Volume 79, Number 269, January 2010, Pages 383–418 S 0025-5718(09)02263-7 Article electronically published on May 12, 2009
THE MINKOWSKI QUESTION MARK FUNCTION: EXPLICIT SERIES FOR THE DYADIC PERIOD FUNCTION AND MOMENTS
GIEDRIUS ALKAUSKAS
Abstract. Previously, several natural integral transforms of the Minkowski question mark function F (x) were introduced by the author. Each of them is uniquely characterized by certain regularity conditions and the functional equation, thus encoding intrinsic information about F (x). One of them, the dyadic period function G(z), was defined as a Stieltjes transform. In this paper we introduce a family of “distributions” Fp(x)for p ≥ 1, such that F1(x)is the question mark function and F2(x) is a discrete distribution with support on x = 1. We prove that the generating function of moments of Fp(x)satisfies the three-term functional equation. This has an independent interest, though our main concern is the information it provides about F (x). This approach yields the following main result: we prove that the dyadic period function is a sum of infinite series of rational functions with rational coefficients.
1. Introduction and main result The aim of this paper is to continue investigations on the moments of the Minkowski ?(x) function, begun in [1], [2] and [3]. The function ?(x) (“the question mark function”) was introduced by Minkowski as an example of a continuous func- tion F :[0, ∞) → [0, 1), which maps rationals to dyadic rationals, and quadratic irrationals to nondyadic rationals. For a nonnegative real x it is defined by the expression
−a0 −(a0+a1) −(a0+a1+a2) (1) F ([a0,a1,a2,a3, ...]) = 1 − 2 +2 − 2 + ...,
where x =[a0,a1,a2,a3, ...] stands for the representation of x by a (regular) con- tinued fraction [15]. By tradition, this function is more often investigated in the interval [0, 1]. Accordingly, we make a convention that ?(x)=2F (x)forx ∈ [0, 1]. For rational x, the series terminates at the last nonzero partial quotient an of the continued fraction. This function is continuous, monotone and singular [9]. A by far not complete overview of the papers written about the Minkowski ques- tion mark function or closely related topics (Farey tree, enumeration of rationals, Stern’s diatomic sequence, various 1-dimensional generalizations and generaliza- tions to higher dimensions, statistics of denominators and Farey intervals, Haus- dorff dimension and analytic properties) can be found in [1]. These works include
Received by the editor September 15, 2008 and, in revised form, January 17, 2009. 2000 Mathematics Subject Classification. Primary 11A55, 26A30, 32A05; Secondary 40A15, 37F50, 11F37. Key words and phrases. The Minkowski question mark function, the dyadic period function, three-term functional equation, analytic theory of continued fractions, Julia sets, the Farey tree.