# The complex sum of digits function and primes

• Volume: 12, Issue: 1, page 133-146
• ISSN: 1246-7405

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## Abstract

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Canonical number systems in the ring of gaussian integers $ℤ\left[i\right]$ are the natural generalization of ordinary $q$-adic number systems to $ℤ\left[i\right]$. 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.

## How to cite

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Thuswaldner, Jörg M.. "The complex sum of digits function and primes." Journal de théorie des nombres de Bordeaux 12.1 (2000): 133-146. <http://eudml.org/doc/248481>.

@article{Thuswaldner2000,
abstract = {Canonical number systems in the ring of gaussian integers $\mathbb \{Z\}[i]$ are the natural generalization of ordinary $q$-adic number systems to $\mathbb \{Z\}[i]$. 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 $\nu _b$ 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.},
author = {Thuswaldner, Jörg M.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {radix representation; Gaussian integers; sum of digits function; Erdős-Kac Theorem},
language = {eng},
number = {1},
pages = {133-146},
publisher = {Université Bordeaux I},
title = {The complex sum of digits function and primes},
url = {http://eudml.org/doc/248481},
volume = {12},
year = {2000},
}

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
PB - Université Bordeaux I
VL - 12
IS - 1
SP - 133
EP - 146
AB - Canonical number systems in the ring of gaussian integers $\mathbb {Z}[i]$ are the natural generalization of ordinary $q$-adic number systems to $\mathbb {Z}[i]$. 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 $\nu _b$ 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.
LA - eng
KW - radix representation; Gaussian integers; sum of digits function; Erdős-Kac Theorem
UR - http://eudml.org/doc/248481
ER -

## References

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