Nous montrons qu'une fonction complètement multiplicative 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 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.
Accepté le :
Publié le :
@article{CRMATH_2019__357_10_752_0, author = {Klurman, Oleksiy and Kurlberg, P\"ar}, title = {A note on multiplicative automatic sequences}, journal = {Comptes Rendus. Math\'ematique}, pages = {752--755}, publisher = {Elsevier}, volume = {357}, number = {10}, year = {2019}, doi = {10.1016/j.crma.2019.10.002}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2019.10.002/} }
TY - JOUR AU - Klurman, Oleksiy AU - Kurlberg, Pär TI - A note on multiplicative automatic sequences JO - Comptes Rendus. Mathématique PY - 2019 SP - 752 EP - 755 VL - 357 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2019.10.002/ DO - 10.1016/j.crma.2019.10.002 LA - en ID - CRMATH_2019__357_10_752_0 ER -
%0 Journal Article %A Klurman, Oleksiy %A Kurlberg, Pär %T A note on multiplicative automatic sequences %J Comptes Rendus. Mathématique %D 2019 %P 752-755 %V 357 %N 10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2019.10.002/ %R 10.1016/j.crma.2019.10.002 %G en %F CRMATH_2019__357_10_752_0
Klurman, Oleksiy; Kurlberg, Pär. A note on multiplicative automatic sequences. Comptes Rendus. Mathématique, Tome 357 (2019) no. 10, pp. 752-755. doi : 10.1016/j.crma.2019.10.002. http://www.numdam.org/articles/10.1016/j.crma.2019.10.002/
[1] Mock characters and the Kronecker symbol, J. Number Theory, Volume 192 (2018), pp. 356-372
[2] Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, UK, 2003
[3] Transcendence of generating functions whose coefficients are multiplicative, Trans. Amer. Math. Soc., Volume 364 (2012) no. 2, pp. 933-959
[4] Ensembles presque periodiques k-reconnaissables, Theor. Comput. Sci., Volume 9 (1979) no. 1, pp. 141-145
[5] Suites algébriques, automates et substitutions, Bull. Soc. Math. Fr., Volume 108 (1980) no. 4, pp. 401-419
[6] Harmonic analysis on the positive rationals. Determination of the group generated by the ratios , Mathematika, Volume 63 (2017) no. 3, pp. 919-943
[7] Artin's conjecture for primitive roots, Q. J. Math. Oxford, Ser. 2, Volume 37 (1986) no. 145, pp. 27-38
[8] Subword complexity and non-automaticity of certain completely multiplicative functions, Adv. Appl. Math., Volume 84 (2017), pp. 73-81
[9] A note on multiplicative automatic sequences, II, 2019 | arXiv
[10] On multiplicative automatic sequences, 2019 | arXiv
[11] On completely multiplicative automatic sequences | arXiv
[12] A criterion for non-automaticity of sequences, J. Integer Seq., Volume 6 (2003) no. 3 (5)
[13] Completely multiplicative automatic functions, Integers, Volume 11 (2011) no. 8, p. A31
[14] Multiplicative functions and k-automatic sequences, J. Théor. Nr. Bordx., Volume 13 (2001) no. 2, pp. 651-658
Cité par Sources :
☆ P.K. was partially supported by the Swedish Research Council (2016-03701).