In Chapter 20 of the book Opera de Cribro, when proving there are infinitely many primes of the form $x^2+y^2$ where $y$ is also a prime, the authors used the Siegel-Walfisz Theorem in the following form:

For any character $\psi$ on ideals we have \begin{equation} \sum_{N\mathfrak{m}\le x}\mu(\mathfrak{m})\psi(\mathfrak{m})\le x(\log x)^{-A}, \end{equation} for any $A>0$, where the implied constant depends only on $A$. Here we are summing over Gaussian integers.

However, usually for this form of Siegel-Walfisz we need the assumption $q\le (\log x)^N$ for some $N>0$ where $q$ is the modulus of the character. But it seems that in the context the modulus can be as large as $x$. Or did I overlook something?


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