I am reading section 12 (Flipping Moduli) of the paper “The polynomial $ X^2+Y^4$ captures its primes” by Friedlander and Iwaniec.

At page 997, just below equation (12.7) we start estimating the following sum $ $ V(f,g)=\sum_d f(d)\sum_{\substack{r_1s_2\equiv r_2s_1 (d)\}} \alpha_{r_1,s_1}\bar{\alpha}_{r_2,s_2}\left(\frac{d}{r_1r_2}\right)g\left(\left|\frac{s_1}{r_1}-\frac{s_2}{r_2}\right|\right) $ $ where $ R\leq r\leq 2R$ , $ S\leq s\leq 2S$ and $ f$ is supported on $ \frac{1}{2}D\leq d\leq 3D$ . They start by reducing the variables $ r_1, r_2$ by the common divisor $ c=(r_1,r_2)$ and removing the resulting condition $ (c,d)=1$ (we know that both $ r_1$ and $ r_2$ are coprime to $ d$ , hence so is their gcd) by Mobius inversion, i.e. $ $ 1_{\text{gcd}(c,d)=1}=\sum_{m|\text{gcd}(c,d)}\mu(m) $ $ Using these tools we obtain equation (12.8) $ $ V(f,g)=\sum_c\sum_{m\mid c}\mu(m)V_{c,m})(f,g) $ $ where $ V_{c,m}(f,g)$ is defined as $ $ V_{c,m}(f,g)=\sum_d f(dm)\sum_{\substack{r_1s_2\equiv r_2s_1 (dm)\ (r_1,r_2)=1}}\alpha_{cr_1,s_1}\bar{\alpha}_{cr_2,s_2}\left(\frac{d}{r_1r_2}\right)g\left(\left|\frac{s_1}{r_1}-\frac{s_2}{r_2}\right|\right) $ $ They then proceed to obtain a bound for $ V_{c,m}(f,g)$ . After two pages they are able to prove the following bound (the inequality after equation (12.16)) \begin{equation} V_{c,m}(f,g)\ll\left\{(cm)^{-1/2}(DHRS)^{1/2}(2\log 2RS)^3 +\left[c^{-3/2}mD^{-1}(RS)^{3/2}+RS^{3/4}+SR^{3/4}\right](RS)^\epsilon\right\}\sum_{r,s}|\alpha_{cr,s}|^2 \end{equation} They then say that summing over this bound over $ m$ and $ c$ as in equation (12.8) yields $ $ V(f,g)\leq \mathcal{H}(D,H,R,S)\sum_r\sum_s\tau(r)|\alpha_{r,s}|^2 $ $ where $ \mathcal{H}(D,H,R,S)$ satisfies the bound \begin{multline} \mathcal{H}(D,H,R,S)\ll (DHRS)^{1/2}(\log 2RS)^4\ +\left[D^{-1}(RS)^{3/2}+RS^{3/4}+SR^{3/4}\right](RS)^\epsilon \end{multline} I am not able to understand how summing over $ m$ and $ c$ yields this bound. In particular what is the range over which $ c$ varies? I think it would suffices to prove the following bounds $ $ \sum_c\sum_{m\mid c}\mu(m)(cm)^{-1/2}\ll \log 2R $ $ and $ $ \sum_c\sum_{m\mid c}\mu(m)(c)^{-3/2}m\ll \log 2R $ $ I am able to show, using the multiplicativity of $ \mu$ that $ $ \sum_{m\mid c}\mu(m)m^{-1/2}\leq 1\qquad\text{ and }\qquad\sum_{m\mid c}\mu(m)m\leq c $ $ but I think that this estimation is too crude. In particular using such bound both equations reduce to showing that $ $ \sum_c c^{-1/2}\ll \log 2R $ $ but doesn’t look to be correct. Overall I don’t know if what I have done so far is correct and I should I proceed now. Thank you!