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 νb, 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 νb un théorème du type Erdös-Kac. Dans ces résultats, l’équidistribution de νb 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 νb 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 νb. In all proofs the equidistribution of νb 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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