Moments of the weighted sum-of-digits function

Abstract

The weighted sum-of-digits function is a slight generalization of the well known sum-of-digits function with the difference that here the digits are weighted by some weights. So for example in this concept also the alternated sum-of-digits function is included. In this paper we compute the first and the second moment of the weighted sum-of-digits function and we draw some consequences of our results. Furthermore we give an alternative representation to Delange's formula for the first moment of the sum-of-digits function. Here we obtain a non-periodic but piece-wise differentiable fluctuation instead of a periodic and nowhere differentiable fluctuation as in Delange's result. Starting from this result we show that a (weak) Delange type result for the first moment of the weighted sum-of-digits function exists iff the sequence of weights converges.

Quaestiones Mathematicae 28(2005), 321–336.