The images of the upper half of the unit circle \(z= e^{i\varphi}\), \(0\leq \varphi\leq \pi\), under several mappings are investigated. These mappings are determined by the power series of the form \(\sum^\infty_{n= 1} f(n) z^n/n\), where \(f(n)\) is a function which may depend on e.g. the structure of the number \(n\). As \(f(n)\) were chosen for example the Möbius function \(\mu(n)\) or \(\{\sqrt n\}\), \(\{\ln n\}\), (where \(\{\cdot\}\) denotes the fractional part), \(\sin n^2\) and several others. The images in question are drawn by computer and exhibit fractal properties. The authors also have tried to determine the box-counting dimensions of the resulting fractal objects.