## 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

Zbl 0447.68083
Full Text:

### References:

 [1] J. P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge, 2003. · Zbl 1086.11015 [2] T. M. Apostol, Introduction to Analytic Number Theory. Springer-Verlag, UTM, 1976. · Zbl 1154.11300 [3] H. Delange, Sur la fonction sommatoire de la fonction “somme des chiffres”. L’Enseignement Math. 21 (1975), 31-47. · Zbl 0306.10005 [4] J. M. Dumont and A. Thomas, Systemes de numeration et fonctions fractales relatifs aux substitutions. Theoretical Computer Science 65 (1989), 153-169. · Zbl 0679.10010 [5] P. Flajolet and L. Ramshaw, A note on Gray code and odd-even merge. SIAM J. Comput. 9 (1980), 142-158. · Zbl 0447.68083 [6] P. Flajolet, P. Grabner, P. Kirschenhofer, H. Prodinger, and R. F. Tichy, Mellin transforms and asymptotics: digital sums. Theoretical Computer Science 123 (1994), 291-314. · Zbl 0788.44004 [7] F. Gray, Pulse Code Communications. U.S. Patent 2632058, March 1953. [8] J. L. Mauclaire and L. Murata, An explicit formula for the average of some $$q$$-additive functions. Proc. Prospects of Math. Sci., World Sci. Pub. (1988), 141-156. · Zbl 0658.10064 [9] C. E. Killian and C. D. Savage, Antipodal Gray Codes. Discrete Math. 281 (2004), 221-236. · Zbl 1054.94020
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