La notion de développement -adique d’un entier, pour une base donnée, se généralise dans l’anneau des entiers de Gauss au développement d’un entier de Gauss suivant une certaine base , ce développement étant unique. Dans cet article, on s’intéresse à la fonction , désignant la somme de chiffres dans le développement suivant la base . On montre un résultat sur la fonction somme de chiffres pour les nombres non multiples d’une puissance -ième d’un nombre premier. On établit aussi pour un théorème du type Erdös-Kac. Dans ces résultats, l’équidistribution de joue un rôle essentiel. Partant de cela, les démonstrations font alors appel à des méthodes de crible, ainsi qu’à une version du modèle de Kubilius.
Canonical number systems in the ring of gaussian integers are the natural generalization of ordinary -adic number systems to . It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number . In this paper we investigate the sum of digits function of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the -th power of a prime. Furthermore, we establish an Erdös-Kac type theorem for . In all proofs the equidistribution of in residue classes plays a crucial rôle. Starting from this fact we use sieve methods and a version of the model of Kubilius to prove our results.
@article{JTNB_2000__12_1_133_0, author = {Thuswaldner, J\"org M.}, title = {The complex sum of digits function and primes}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {133--146}, publisher = {Universit\'e Bordeaux I}, volume = {12}, number = {1}, year = {2000}, mrnumber = {1827844}, zbl = {1012.11071}, language = {en}, url = {http://archive.numdam.org/item/JTNB_2000__12_1_133_0/} }
TY - JOUR AU - Thuswaldner, Jörg M. TI - The complex sum of digits function and primes JO - Journal de théorie des nombres de Bordeaux PY - 2000 SP - 133 EP - 146 VL - 12 IS - 1 PB - Université Bordeaux I UR - http://archive.numdam.org/item/JTNB_2000__12_1_133_0/ LA - en ID - JTNB_2000__12_1_133_0 ER -
Thuswaldner, Jörg M. The complex sum of digits function and primes. Journal de théorie des nombres de Bordeaux, Tome 12 (2000) no. 1, pp. 133-146. http://archive.numdam.org/item/JTNB_2000__12_1_133_0/
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