The complex sum of digits function and primes

Thuswaldner, Jörg M.

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 $\mathbb{Z}\left[i\right]$ au développement d’un entier de Gauss suivant une certaine base $b\in \mathbb{Z}\left[i\right]$, ce développement étant unique. Dans cet article, on s’intéresse à la fonction ${\nu }_b$, 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 ${\nu }_b$ un théorème du type Erdös-Kac. Dans ces résultats, l’équidistribution de ${\nu }_b$ 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 $\mathbb{Z}\left[i\right]$ are the natural generalization of ordinary $q$-adic number systems to $\mathbb{Z}\left[i\right]$. 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 ${\nu }_b$ 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 ${\nu }_b$. In all proofs the equidistribution of ${\nu }_b$ 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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