Multiplicative functions and $k$-automatic sequences
Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 2, p. 651-658

A sequence is called $k$-automatic if the $n$’th term in the sequence can be generated by a finite state machine, reading $n$ in base $k$ as input. We show that for many multiplicative functions, the sequence ${\left(f\left(n\right)\phantom{\rule{4pt}{0ex}}\text{mod}\phantom{\rule{4pt}{0ex}}v\right)}_{n\ge 1}$ is not $k$-automatic. Among these multiplicative functions are ${\gamma }_{m}\left(n\right),{\sigma }_{m}\left(n\right),\mu \left(n\right)$ et $\phi \left(n\right)$.

Une suite est dite $k$-automatique si son ${n}^{e}$ terme peut être engendré par une machine à états finis lisant en entrée le développement de $n$ en base $k$. Nous prouvons que, pour de nombreuses fonctions multiplicatives $f$, la suite ${\left(f\left(n\right)\phantom{\rule{4pt}{0ex}}\text{mod}\phantom{\rule{4pt}{0ex}}v\right)}_{n\ge 1}$ n’est pas $k$-automatique. C’est en particulier le cas pour les fonctions multiplicatives ${\gamma }_{m}\left(n\right),{\sigma }_{m}\left(n\right),\mu \left(n\right)$ et $\phi \left(n\right)$.

@article{JTNB_2001__13_2_651_0,
author = {Yazdani, Soroosh},
title = {Multiplicative functions and $k$-automatic sequences},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Universit\'e Bordeaux I},
volume = {13},
number = {2},
year = {2001},
pages = {651-658},
zbl = {1002.11025},
mrnumber = {1879677},
language = {en},
url = {http://www.numdam.org/item/JTNB_2001__13_2_651_0}
}

Yazdani, Soroosh. Multiplicative functions and $k$-automatic sequences. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 651-658. http://www.numdam.org/item/JTNB_2001__13_2_651_0/

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