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

It is known, that under the assumption of the generalized Riemannian hypothesis, the function $π(x, q, 1) - π(x, q, a)$ 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 $π(x)-\li x$. 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 $ζ$-function.

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