I am trying to find an estimate of the following sum:
$$S(P)=\sum_i \int_{[0,\frac{\tau}{P}]^{n-1}} \left(\frac{\operatorname{li}(\tau)}{\tau}\right)^{n-1} \int_{[\frac{\tau}{P},1]} \frac{e(\gamma F(z_i))}{\log(Pz_i)} \, \mathrm{d}{z_i}\, \mathrm{d}\hat{z_i},$$ where $\tau\ge2$ and $P$ are positive integers, $e(x)=e^{2\pi ix}$, and $F$ is a homogeneous polynomial in $n$-variables and $i\in\{1,2,\dots,n\}$. I want this sum to be $S(P)\sim h(P)$. (Note $\mathrm{d}\hat{z_i}=\mathrm{d}z_1\dots\mathrm{d}z_n$ except $\mathrm{d}z_i$.)
What I have tried: I know that a complex-valued function $f(z)=O(g(z))$ in an open subset $U\subset\mathbb{C}$, then $|f(z)|\leq M|g(z)|$ for all $z\in\overline{U}$ for a constant $M>0$. Then, if I just consider one particular integral inside the sum and take its absolute value, from the triangle inequality we obtain $$\bigg|\int_{[0,\frac{\tau}{P}]^{n-1}} \left(\frac{\operatorname{li}(\tau)}{\tau}\right)^{n-1} \int_{[\frac{\tau}{P},1]}\frac{e(\gamma F(z_i))}{\log(Pz_i)}\, \mathrm{d}{z_i}\, \mathrm{d}\hat{z_i}\bigg|\leq \left( \frac{\operatorname{li}(\tau)} \tau \right)^{n-1} \int_{[0,\frac{\tau}{P}]^{n-1}} \int_{[\frac{\tau}{P},1]} \bigg| \frac{e(\gamma F(z_i))}{\log(Pz_i)}\bigg|\, \mathrm{d}{z_i}\, \mathrm{d}\hat{z_i}.$$
Question Is this correct? If yes, then would it be okay to also estimate the sum $$\sum_{i,j}\int_{[0,\frac{\tau}{P}]^{n-2}}\left(\frac{\operatorname{li}(\tau)}{\tau}\right)^{n-2}\int_{[\frac{\tau}{P},1]^2}\frac{e(\gamma F(z_i,z_j))}{\log(Pz_i)\log(Pz_j)}\, \mathrm{d}{z_i}\, \mathrm{d}z_j\, \mathrm{d}\hat{z}$$ similarly? If not, then I would really appreciate it if somebody could show me how to find these estimates.
