# On the Moments of the Sum-of-Digits Function

**
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.

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

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

Peter.Kirschenhofer@tuwien.ac.at,

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

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