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Relations among arithmetical functions, automatic sequences, and sum of digits functions induced by certain Gray codes. (English. French summary) Zbl 1280.11017

Authors’ summary: “In the study of the 2-adic sum of digits function \(S_2(n)\), the arithmetical function \(u(0)=0\) and \(u(n)=(-1)^{n-1}\) for \(n\geq 1\) plays a very important role. In this paper, we firstly generalize the relation between \(S_2(n)\) and \(u(n)\) to a bijective relation between arithmetical functions. And as an application, we investigate some aspects of the sum of digits functions \(S_{\mathcal G}(n)\) induced by binary infinite Gray codes \(\mathcal G\). We can show that the difference of the sum of digits function, \(S_{\mathcal G}(n)- S_{\mathcal G}(n-1)\), is realized by an automaton. And the summation formula of the sum of digits function for reflected binary code, proved by P. Flajolet and L. Ramshaw [SIAM J. Comput. 9, 142–158 (1980; Zbl 0447.68083)], is also generalized. Here we use analytic tools such as Mellin transform and Perron’s formula for Dirichlet series.”
Note added by the reviewer: The authors have recently obtained more general results [J. Théor. Nombres Bordx. 27, No. 1, 149–169 (2015; 06554400)].

MSC:

11B85 Automata sequences
11A25 Arithmetic functions; related numbers; inversion formulas
11A63 Radix representation; digital problems

Citations:

Zbl 0447.68083
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References:

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