# Mellin Transforms and Asymptotics:
Digital Sums

**Abstract.
Arithmetic functions related to number representation systems
exhibit various periodicity phenomena.
For instance, a well known theorem of Delange expresses the
total number of ones in the binary representations of
the first n integers
in terms of a periodic fractal function.
We show that such periodicity phenomena can be analyzed rather systematically
using classical tools from analytic number theory, namely
the Mellin-Perron formulae.
This approach yields naturally
the Fourier series involved in
the expansions of a variety of digital sums related to number representation
systems.
**

helmut@gauss.cam.wits.ac.za
Here are the addresses of my coauthors:

Philippe.Flajolet@inria.fr,

grabner@weyl.math.tu-graz.ac.at,

Peter.Kirschenhofer@tuwien.ac.at,

tichy@weyl.math.tu-graz.ac.at,

This paper is available in the Tex, Dvi, and PostScript format.

If you go to the homepage of
Philippe Flajolet,
you will find, amongst a lot of interesting things, a postscript version of this paper,
which **includes** the graphics!

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