Comptes Rendus
no. 10
Number theory/Combinatorics
A note on multiplicative automatic sequences
[Une note sur les séquences multiplicatives automatiques]

Présenté par : the Editorial Board

Oleksiy Klurman1 ; Pär Kurlberg1
1 Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden
Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 752-755.

Nous montrons qu'une fonction complètement multiplicative f:N→C qui est également q-automatique est ultimement périodique, coïncidant donc avec un caractère de Dirichlet pour tout nombre premier suffisamment grand. Ceci résout un problème de J.-P. Allouche et L. Goldmakher et confirme une conjecture de J. Bell, N. Bruin et M. Coons pour les fonctions complètement multiplicatives. De plus, sous l'hypothèse de Riemann généralisée, notre démonstration peut être adaptée pour prouver cette conjecture pour toute fonction simplement supposée multiplicative.

We prove that any q-automatic completely multiplicative function f:N→C essentially coincides with a Dirichlet character. This answers a question of J.-P. Allouche and L. Goldmakher and confirms a conjecture of J. Bell, N. Bruin and M. Coons for completely multiplicative functions. Further, assuming GRH, the methods allow us to replace completely multiplicative functions with multiplicative functions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.10.002
Oleksiy Klurman 1 ; Pär Kurlberg 1

1 Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden 
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Oleksiy Klurman; Pär Kurlberg. A note on multiplicative automatic sequences. Comptes Rendus. Mathématique, Volume 357 (2019) no. 10, pp. 752-755. doi : 10.1016/j.crma.2019.10.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.10.002/

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☆ P.K. was partially supported by the Swedish Research Council (2016-03701).

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