# range of sum in “the polynomial $X^2+Y^4$ captures its prime”

I am reading section 12 (Flipping Moduli) of the paper "The polynomial $$X^2+Y^4$$ captures its primes".

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 $$\sum_{m\mid c}\mu(m)m^{-1/2}\leq 1\qquad\text{ and }\qquad\sum_{m\mid c}\mu(m)m\leq c$$ Using this both equations reduce to showing that $$\sum_c c^{-1/2}\ll \log 2R$$ but I don't know if what I have done so far is correct and I should I proceed now. Thank you!