On a system of step functions

It is well-known that the Riemann hypothesis is equivalent to the assertion that the identity function belongs to the linear span in $L^2(0,1)$ of the following function set
\begin{equation} \left[\frac\alpha x\right]-\alpha\left[\frac1x\right], \qquad 0<\alpha<1. \tag{1} \end{equation}
A step is presented in describing the set of all idempotents representable as a finite sum of functions of the form (1).