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.


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