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\leftheadtext{P.J. Grabner,
P. Kirschenhofer, H. Prodinger and
R.F. Tichy}
\define\({\left(}
\define\){\right)}
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\topmatter
\title On the Moments of the Sum-of-Digits Function \endtitle
\author P.J. Grabner$^\dag$,
P. Kirschenhofer, H. Prodinger and
R.F. Tichy$^\dag$\endauthor
\thanks
{$^\dag$ These authors were
supported by the Austrian Science Foundation (Project Nr. P8274PHY)}
\endthanks
\address
\hbox{
\vbox{\hbox to 6 truecm {P. Grabner and R. Tichy\hfill}
\hbox to 6 truecm{Institut f\"ur Mathematik\hfill}
\hbox to 6 truecm{TU Graz\hfill}
\hbox to 6 truecm{Steyrergasse 30\hfill}
\hbox to 6 truecm{8010 Graz, Austria\hfill}}
\vbox{\hbox to 6 truecm{P. Kirschenhofer and H. Prodinger\hfill}
\hbox to 6 truecm{Institut f\"ur Algebra und\hfill}
\hbox to 6 truecm{Diskrete Mathematik, TU Wien\hfill}
\hbox to 6 truecm{Wiedner Hauptstra\ss e 8--10\hfill}
\hbox to 6 truecm{1040 Wien, Austria\hfill}}}
\endaddress
\abstract We are concerned
with the asymptotic description of the higher
moments of the sum-of-digits function.
For the mean value a completely satisfactory result
involving a continuous and no-where differentiable oscillation
function is due to Trollope (1968) and
H. Delange (1975). The asymptotic behaviour of the second moment of the
binary sum-of-digits
function was investigated by J. Cocquet (1986) and by
P. Kirschenhofer (1990). A somewhat weaker result in the decimal number
system is due to R. E. Kennedy and C. N. Cooper (1991).
In the present paper we prove similar (more
or less explicit) formulas in the case of $s$-th moments by a generalization of Delange`s
approach. Furthermore we discuss a second method
for proving such results: the so-called Mellin-Perron
summation formula.
\endabstract
\endtopmatter
\heading 1. Introduction\endheading
In a recent paper in the {\sl Fibonacci Quarterly\/}
R. E. Kennedy and C. N. Cooper \cite{K-C91} dealt with the second moment
of the sum-of-digits function in the decimal number system, explicitely
stating the question for higher moments as an open problem. We consider
this question in the sequel.
Let $\nu(n)$ represent the total number of $1$-digits in the binary
representation
of the integer $n$. It is not hard to see that
$$M_1(N)=\sum_{n