Abstract. The aim of the present paper is the estimation of the $d$th moment of additive functions in canonical number systems. These number systems are generalizations of the decimal number system to arbitrary polynomials having integer coefficients. We call a function additive (with respect to a number system) if it only acts on the digits of an expansion. The sum-of-digits function, as a special additive function, has been analyzed in the case of $q$-adic number systems by Delange and number systems in number fields by Gittenberger and Thuswaldner. The present paper is a generalization of these results to arbitrary additive functions in canonical number systems.

AMS Subject Classification (1991): 11K16, 11R47

Keyword(s): canonical number systems, $b$-additive functions, moment function

Received January 23, 2012, and in revised form April 11, 2012. (Registered under 5/2012.)