Given this function defined on $\mathbb{Z}$:
$f(k)=\left\{\begin{array}{c} 1 \text{, if $(\alpha k+\beta$ mod $\gamma) < \alpha$} \\ 0 \text{, otherwise}\end{array} \right\}$
I want to calculate:
$S = \sum_{k=a}^{b} f(k)$
where $a, b \in \mathbb{Z}$ and $b > a$.
It's quite easy to see that $S$ sometimes is the integer part of $(b-a)\frac{\alpha}{\gamma}$ and sometimes is the ceiling part of $(b-a)\frac{\alpha}{\gamma}$.
Is there a way to calculate $S$ precisely and directly? Furthermore, given $\alpha, \gamma, a, b$ how does $S$ depend on $\beta$?
Thanks in advance.