Sign changes of $\pi (x,q,1)-\pi (x,q,a)$

It is known, that under the assumption of the generalized Riemannian hypothesis, the function $\pi (x,q,1)-\pi (x,q,a)$ has infinitely many sign changes. In this article we give an upper bound for the least such sign change. Similarly, assuming the Riemannian hypothesis we give a lower bound for the number of sign changes of $\pi (x)-\text{li}x$. The implied results for the least sign change are weaker then those obtained by numerical methods, however, our method makes no use of computations of zeros of the $\zeta$-function.