I have a question about the argumentation at the beginning of section 15 in this paper. The goal is to estimate the sum 
$$V(\beta) = 2 \sum\_{(z\_1,z\_2)=1} \beta\_{z\_1} \overline{\beta\_{z\_2}} \sum\_{\substack{de > X \\ r\_1s\_2 \equiv r\_2s\_1 \text{ } (4d)}} \frac{\varphi(de)}{de} f \Big( \frac{|\Delta|}{de} \Big) \left(\dfrac{s\_1s\_2}{e}\right) \left(\dfrac{r\_1r\_2}{d}\right) \log \Big(2 \big| \frac{z\_1z\_2}{\Delta} \big| \Big),$$
where the summation is restricted to those $z\_1,z\_2 \in \mathbb{Z}[i]$ fulfilling
$$z\_1 \equiv z\_2 \equiv z\_0 \text{ (mod } 8 \text{)},$$
where $z\_0$ is assumed to be primary, i.e., $z\_0 \equiv 1$ (mod $2(i+1)$), and
$$(z\_1, \prod\_{p \leq P} p) = (z\_2, \prod\_{p \leq P} p) = 1,$$
where $P$ is a constant to be specified later. Here, we write $z\_j = e r\_j + i s\_j$ for $j=1,2$.


Further, $\Delta = \Delta(z\_1,z\_2) = \mathrm{Im}(z\_1 \overline{z\_2}) = \frac{1}{2i}(\overline{z\_1}z\_2-z\_1 \overline{z\_2})$, $|\Delta| \leq N$ and $f$ is a function that can be bounded (in absolute value) by $1$. Moreover, we define
$$\beta\_z = q(\mathrm{arg}(z))p(z \overline{z}) \mu(z \overline{z}) \sum\_{c \mid z \overline{z}, c \leq C} \mu(c),$$
where $q$ is a $2\pi$-periodic $C^2$-function supported on $\phi < \mathrm{arg}(z) \leq \phi + 2 \pi \theta$ such that $q^{(j)} \ll \theta^{-j}$ for $j=0,1,2$, and $p$ is a twice differentiable (or even smooth) function supported on $N' \leq n \leq N'(1+\theta)$ such that $p^{(j)} \ll (\theta N)^{-j}$ for $j=0,1,2$, where $N < N' < 2N$, and $\theta^{-1}$ will be chosen to be a large power of $\log(N)$.


In particular, $\beta\_z$ is supported on the polar box $$\{ z \mid N' < |z|^2 \leq (1+\theta)N', \phi < \mathrm{arg}(z) \leq \phi + 2 \pi \theta \}$$
of volume $\pi \theta^2 N' \ll \theta^2 N$.


Now they claim that the condition $(z\_1,z\_2)=1$ can be dropped at a cost of $\mathcal{O}(N^2P^{-1})$. Moreover: "To get it at this cost, apply Lemma 2.2 with respect to $k=4$ to reduce $d$ before estimating trivially."


(The relevant part of) Lemma 2.2 says that, for any fixed $k \geq 2$, any $n \geq 1$ has a divisor $d \leq n^{1/k}$ such that
$$\tau(n) \leq (2 \tau(d))^{\frac{k \log(k)}{\log(2)}}.$$


I guess that they want to consider the sum $2 \sum\_{(z\_1,z\_2) \neq 1} ...$, a similar argumentation also appears between the equations (5.9) and (5.10) in their paper. Further, it is clear that the terms involving $\phi$, $f$ or the Legendre symbols can be bounded by $1$, and the $\log$-term can be bounded by $\log(N)$ (which can be compensated later, since $\theta^{-1}$ is a large power of $\log(N)$).


However, this is the point where I get stuck. I do not understand what they mean by "reducing $d$", I am not sure how to estimate the $\beta\_z$, and I do not know how the volume of the polar box and the fact that  

$$(z\_1, \prod\_{p \leq P} p) = (z\_2, \prod\_{p \leq P} p) = 1$$
come into play to finally get their desired bound. Could anyone please help me?



Question about Friedlander, Iwaniec: "The polynomial $X^2+Y^4$ captures its primes"
Note first the $\beta\_z$ are bounded by $\tau(|z|^2)$.


What you then have as a bound with removing the gcd in 15.2 is a sum over $g=\gcd(z\_1,z\_2)$ as
$$\sum\_{|g|\ge P}\tau(g)^2\sum\_{|z\_i|\le N/|g|\atop \text{$z\_i$ in box}}\tau(|z\_1|^2)\tau(|z\_2|^2)\sum\_{e|\gcd({\rm Re}(z\_1),{\rm Re}(z\_2))}\sum\_{d\le N\atop g^2\Delta(z\_1,z\_2)\equiv 0 (4d)}(\log N)$$


The innermost log-term comes from (7.3), everything else being bounded by 1. Note that we've saved essentially $P^2$ from the sums over both $z\_1$ and $z\_2$ being smaller by that factor, though we lose back one $P$ from the $g$-sum (see below). See 4.4 and perhaps 5.24 and 4.11 for the size of $P$ relative to everything else.


The remaining $d$-sum can be taken over the divisors of $g^2\Delta$, and similarly with the $e$-sum over the relevant divisors with the real parts of $z\_i$.


Now we can argue as with (10.8) and (10.9) and its usage of Lemma 2.2 (though perhaps with a different letter-name for $d$). This introduces some large amount of log factors.


As they say, the box factor (with $\theta$) saves more logarithms than we lose from the arithmetics, and we are reduced to something like
$$\theta^2N^2(\log N)^{\text{big}}\sum\_{|g|\ge P} \frac{\tau(g)^4}{g^2}$$
which indeed saves us the $P$ we desire.


