The following paragraph appears in Analytic Number Theory (Iwaniec, Kowalski):




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The Siegel-Walfisz theorem asserts that:


$\displaystyle \hspace{5cm} \psi(x;q,a) = \frac{x}{\phi(q)} + O(x(\log x)^{-A})$


for any $q\geq 1, (a,q)=1, x\geq 2$ and $A\geq 0$. **Notice that this estimate is non-trivial only if $q \ll (\log x)^A$**.




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The last sentence is somewhat clear to me intuitively, and ought to answer my question. But I am not quite sure what Vinogradov's '$\ll$' notation is taken to mean in this context, as $q$ is not even a function of $x$. Can anyone clarify this?



When is the Siegel-Walfisz theorem non-trivial?
The $\ll$ in this context means that there is some constant $C > 0$ such that $q \leq C (\log{x})^A$. If $q$ goes to infinity (with $x$) much faster than this then the main term itself can be bounded by the error term (in which case the result would be "trivial").


