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In the study of the 2-adic sum of digits function S2(n), the arithmetical function u(0) = 0 and u(n) = (−1)n−1 for n ≥ 1 plays a very important role. In this paper, we firstly generalize the relation between S2(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 SG(n) induced by binary infinite Gray codes G. We can show that the difference of the sum of digits function, SG(n) − SG(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. Manuscrit reçu le 1er février 2011. Mots clefs. arithmetical function, sum of digits function, Gray code, automatic sequence, Delange’s theorem. Classification math. 11A25, 11B85. 308 Yuichi Kamiya, Leo Murata

Relations among arithmetical functions, automatic sequences, and sum of digits functions induced by certain Gray codes