Relations among arithmetical functions, automatic sequences, and sum of digits functions induced by certain Gray codes
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 2, pp. 307-337.

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 n1 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 𝒢 (n) induced by binary infinite Gray codes 𝒢. We can show that the difference of the sum of digits function, S 𝒢 (n)-S 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, is also generalized. Here we use analytic tools such as Mellin transform and Perron’s formula for Dirichlet series.

Dans l’étude de la somme des chiffres S 2 en base deux, la fonction arithmétique u définie par u(0)=0 et u(n)=(-1) n-1 pour n1 joue un rôle de première importance. Dans cet article, nous commençons par généraliser la relation entre S 2 et u en introduisant une permutation sur l’ensemble des suites à valeurs complexes, nulles en 0. Comme application, certaines relations impliquant la fonction somme des chiffres S 𝒢 associée à un code binaire infini 𝒢 de type Gray sont mises en vidence. En particulier nous montrons que la différence nS 𝒢 (n)-S 𝒢 (n-1) s’obtient par un automate. La formule sommatoire de P. Flajolet et L. Ramshaw pour la somme des chiffres associée au classique code refléchi de Gray est aussi généralisée. La méthode est analytique ; elle utilise la tranformée de Mellin et la formule de Perron pour les séries de Dirichlet.

DOI: 10.5802/jtnb.798
Classification: 11A25,  11B85
Keywords: arithmetical function, sum of digits function, Gray code, automatic sequence, Delange’s theorem.
Kamiya, Yuichi 1; Murata, Leo 2

1 Department of Modern Economics Faculty of Economics Daito Bunka University 560 Iwadono, Higashi-Matsuyama Saitama 355-8501, Japan
2 Department of Mathematics Faculty of Economics Meiji Gakuin University 1-2-37 Shirokanedai Minato-ku, Tokyo 108-8636, Japan
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Kamiya, Yuichi; Murata, Leo. Relations among arithmetical functions, automatic sequences, and sum of digits functions induced by certain Gray codes. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 2, pp. 307-337. doi : 10.5802/jtnb.798. http://www.numdam.org/articles/10.5802/jtnb.798/

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