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The sum-of-digits function for complex bases. (English) Zbl 0959.11045

Given a fixed positive integer \(a\), Gaussian integers admit a unique \((-a+i)\)-ary representation as \(\sum_{l=0}^k \varepsilon_l (-a+i)^l\), where the digits belong to \(\{0,\dots, a^2\}\). The aim of this paper is to study the sum-of-digits function for this numeration system. After recalling some basic facts on these digital expansions, a finite automaton is produced performing the addition of a fixed Gaussian integer. Results on the length of these digital expansions are then derived in connection with the fundamental region \(\{ \sum_{l=0}^{+\infty} \varepsilon_l (-a+i)^{-l}\}\). An asymptotic formula for the sum-of-digits functions in large circles is then expressed. The proof is based on Delange’s approach to the summatory function in the \(q\)-ary case [H. Delange, Enseign. Math., II. Ser. 21, 31-47 (1975; Zbl 0306.10005)]. Moreover, the sum-of-digits function is proved to be uniformly distributed with respect to the argument, by applying the Mellin-Perron summation formula for Dirichlet generating functions.
This paper ends with the study of the summatory function of the sum-of-digits function along the real axis. Note that these results have been extended by J. M. Thuswaldner [Bull. Lond. Math. Soc. 30, 37-45 (1998; Zbl 0921.11051)] to more general number fields.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11A63 Radix representation; digital problems
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
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