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


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 $q \ll (\log x)^A$.


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?

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up vote 6 down vote accepted

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").

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OK, I understand this a little better. But your sentence "If $q$ goes to infinity (with $x$)" seems to imply that we tend to work with $q$ that changes with $x$. Is this the case? Because if $q$ is constant, surely $q \leq C(\log x)^A$ is <i>always</i> going to be true for some $C>0$. –  Sputnik Jan 16 '11 at 14:12
    
Right, people are often most interested in how big $q$ can be allowed to be compared to $x$. If $q$ is completely fixed then everything is on the same level of difficulty as the prime number theorem. –  Matt Young Jan 16 '11 at 18:42
    
Fahad, the first application of this theorem is Vinogradov's proof that large odd numbers have (many) representations as a sum of three primes. The proof uses the circle method which makes it necessary to understand the behavior of exponential sums around rational numbers with small denominators $q$ (compared to the $N$ you want to represent). This is directly related to the distribution of primes in all arithmetic progressions modulo $q$, where $q$ can be as large as $(\log N)^A$. Rumor(?) says that Vinogradov had the argument 10 years before Siegel-Walfisz, but waited until it appeared! –  GH from MO Jan 16 '11 at 18:59
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