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Average growth-behavior and distribution properties of generalized weighted digit-block-counting functions

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Abstract.

We introduce a generalized weighted digit-block-counting function on the nonnegative integers, which is a generalization of many digit-depending functions as, for example, the well known sum-of-digits function. A formula for the first moment of the sum-of-digits function has been given by Delange in 1972. In the first part of this paper we provide a compact formula for the first moment of the generalized weighted digit-block-counting function and show that a (weak) Delange type formula holds if the sequence of weights converges. The question, whether the converse is true as well, can only be answered partially at the moment.

In the second part of this paper we study distribution properties of generalized weighted digit-block-counting sequences and their d-dimensional analogues. We give an if and only if condition under which such sequences are uniformly distributed modulo one.

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Authors and Affiliations

  1. Universität Linz, Linz, Austria

    Roswitha Hofer, Gerhard Larcher & Friedrich Pillichshammer

Authors
  1. Roswitha Hofer
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  2. Gerhard Larcher
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  3. Friedrich Pillichshammer
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Additional information

Dedicated to Prof. Robert F. Tichy on the occasion of his 50th birthday

Roswitha Hofer, Recipient of a DOC-FFORTE-fellowship of the Austrian Academy of Sciences at the Institute of Financial Mathematics at the University of Linz (Austria).

Friedrich Pillichshammer, Supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

Authors’ address: Roswitha Hofer, Gerhard Larcher and Friedrich Pillichshammer, Institut für Finanzmathematik, Universität Linz, Altenbergerstraße 69, A-4040 Linz, Austria

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Hofer, R., Larcher, G. & Pillichshammer, F. Average growth-behavior and distribution properties of generalized weighted digit-block-counting functions. Monatsh Math 154, 199–230 (2008). https://doi.org/10.1007/s00605-007-0513-1

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  • DOI: https://doi.org/10.1007/s00605-007-0513-1

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