Denote by Liouville’s function concerning the parity of the number of prime divisors of . Using a theorem of Allouche, Mendès France, and Peyrière and many classical results from the theory of the distribution of prime numbers, we prove that is not –automatic for any . This yields that is transcendental over for any prime . Similar results are proven (or reproven) for many common number–theoretic functions, including , , , , , and others.
Soit la fonction de Liouville indiquant la parité du nombre de facteurs dans la décomposition de en facteurs premiers. En combinant un théorème d’Allouche, Mendès France, et Peyrière avec quelques résultats classiques de la théorie de la distribution des nombres premiers, nous démontrons que la fonction n’est pas –automatique pour . Cela entraine que est transcendant sur pour tous les nombres premiers . Nous montrons (ou redémontrons) des résultats semblables pour les fonctions numériques , , , , , et autres fonctions.
@article{JTNB_2010__22_2_339_0, author = {Michael Coons}, title = {(Non)Automaticity of number theoretic functions}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {339--352}, publisher = {Universit\'e Bordeaux 1}, volume = {22}, number = {2}, year = {2010}, doi = {10.5802/jtnb.718}, mrnumber = {2769065}, zbl = {1223.11115}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.718/} }
TY - JOUR AU - Michael Coons TI - (Non)Automaticity of number theoretic functions JO - Journal de théorie des nombres de Bordeaux PY - 2010 SP - 339 EP - 352 VL - 22 IS - 2 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.718/ DO - 10.5802/jtnb.718 LA - en ID - JTNB_2010__22_2_339_0 ER -
%0 Journal Article %A Michael Coons %T (Non)Automaticity of number theoretic functions %J Journal de théorie des nombres de Bordeaux %D 2010 %P 339-352 %V 22 %N 2 %I Université Bordeaux 1 %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.718/ %R 10.5802/jtnb.718 %G en %F JTNB_2010__22_2_339_0
Michael Coons. (Non)Automaticity of number theoretic functions. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 339-352. doi : 10.5802/jtnb.718. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.718/
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