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

Kamiya, Yuichi, and 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 24.2 (2012): 307-337. <http://eudml.org/doc/251032>.

@article{Kamiya2012,
abstract = {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 $n\ge 1$ 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, is also generalized. Here we use analytic tools such as Mellin transform and Perron’s formula for Dirichlet series.},
affiliation = {Department of Modern Economics Faculty of Economics Daito Bunka University 560 Iwadono, Higashi-Matsuyama Saitama 355-8501, Japan; Department of Mathematics Faculty of Economics Meiji Gakuin University 1-2-37 Shirokanedai Minato-ku, Tokyo 108-8636, Japan},
author = {Kamiya, Yuichi, Murata, Leo},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {arithmetical function; sum of digits function; Gray code; automatic sequence; Delange’s theorem; arithmetic function; sum of digits; automatic sequences; Delange theorem},
language = {eng},
month = {6},
number = {2},
pages = {307-337},
publisher = {Société Arithmétique de Bordeaux},
title = {Relations among arithmetical functions, automatic sequences, and sum of digits functions induced by certain Gray codes},
url = {http://eudml.org/doc/251032},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Kamiya, Yuichi
AU - Murata, Leo
TI - Relations among arithmetical functions, automatic sequences, and sum of digits functions induced by certain Gray codes
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/6//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 2
SP - 307
EP - 337
AB - 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 $n\ge 1$ 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, is also generalized. Here we use analytic tools such as Mellin transform and Perron’s formula for Dirichlet series.
LA - eng
KW - arithmetical function; sum of digits function; Gray code; automatic sequence; Delange’s theorem; arithmetic function; sum of digits; automatic sequences; Delange theorem
UR - http://eudml.org/doc/251032
ER -