  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 $\pi(x)-\li x$. 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.
