The complex sum of digits function and primes
Journal de théorie des nombres de Bordeaux, Tome 12 (2000) no. 1, pp. 133-146

Canonical number systems in the ring of gaussian integers ℤ[i] are the natural generalization of ordinary q-adic number systems to ℤ[i]. It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number b. In this paper we investigate the sum of digits function ν b of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the f-th power of a prime. Furthermore, we establish an Erdös-Kac type theorem for ν b . In all proofs the equidistribution of ν b 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.

La notion de développement q-adique d’un entier, pour une base q donnée, se généralise dans l’anneau des entiers de Gauss ℤ[i] au développement d’un entier de Gauss suivant une certaine base b∈ℤ[i], ce développement étant unique. Dans cet article, on s’intéresse à la fonction ν b , désignant la somme de chiffres dans le développement suivant la base b. On montre un résultat sur la fonction somme de chiffres pour les nombres non multiples d’une puissance f-ième d’un nombre premier. On établit aussi pour ν b un théorème du type Erdös-Kac. Dans ces résultats, l’équidistribution de ν b 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.

Jörg M. Thuswaldner. The complex sum of digits function and primes. Journal de théorie des nombres de Bordeaux, Tome 12 (2000) no. 1, pp. 133-146. doi: 10.5802/jtnb.271
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