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

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?

• The amount of different notations per volume unit is amazing! I must admit you spent a lot of time explaining most of them, but some are still unclear. What is $X$? Wat is $C$? What is $N$? Did I miss any others? – Vincent Jan 8 '19 at 10:42
• @Vincent: They are all constants. $N$ first appears in equation (4.3) in the definition of a certain bilinear form $B(x;N)$. $P$ and $N$ seem to depend on $x$ (not to be confused with $X$), where $x$ is the major variable (and is explained in equation (4.2)). Further, $C$ is specified in Prop. 4.1, and $X \leq N^{1/9}$ by Prop. 15.1. However, the problem is that sometimes variables change their meaning during the text. Here, I am refering to this version of the paper: arxiv.org/pdf/math/9811185v1.pdf – Algebrus Jan 8 '19 at 10:55
• I agree with Vincent, unless one already knows that paper's structure, it's hard to parse this surfeit of notation in the context of a self-contained MO question. – literature-searcher Jan 8 '19 at 12:26

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.