Question about Friedlander, Iwaniec: "The polynomial $X^2+Y^4$ captures its primes"

Question

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?

Answer 1

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.