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

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.

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.

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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. https://jtnb.centre-mersenne.org/item/JTNB_2000__12_1_133_0/

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