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\(\alpha\)-expansions, linear recurrences, and the sum-of-digits function. (English) Zbl 0725.11005

The authors give a generalization of the ordinary \(q\)-ary digit expansion of positive integers to arbitrary real base \(\alpha >1\) and investigate the properties of the sum-of-digits function \(s(n)\) with respect to this expansion. An asymptotic formula for the sum \(\sum_{n<N}S(n)\) is given and distribution properties of the sequence \(x\cdot s(n)\) for irrational \(x\) are considered.

MSC:

11A63 Radix representation; digital problems
11N37 Asymptotic results on arithmetic functions
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References:

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