Moments of the weighted sum-of-digits function

  • G Larcher Institut f&#252r Finanzmathematik, Universit&#228t Linz and Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergstraße 69, A-4040 Linz, Austria
  • F Pillichshammer Institut f&#252r Finanzmathematik, Universit&#228t Linz, Altenbergstraße 69, A-4040 Linz, Austria.
Keywords: sum-of-digits function, moments, Delange formula.mage


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.

Journal Identifiers

eISSN: 1727-933X
print ISSN: 1607-3606