Let $c>0$ a real number, let $N$ a large natural number and let $e\left(x\right):=e^{2\pi ix}$. Is it true that $\forall k\in\left[1,\,2N\right]$, $k$ natural number, that $$\left|\underset{\underset{n\_{1}+n\_{2}=k}{2\leq n\_{1},\, n\_{2}\leq N}}{\sum}\left(\log n\_{1}\log n\_{2}\, e\left(c\left(n\_{1}+n\_{2}\right)\right)\right)\right|\leq\left|\underset{2\leq n\_{1},\, n\_{2}\leq N}{\sum}\left(\log n\_{1}\log n\_{2}\,e\left(c\left(n\_{1}+n\_{2}\right)\right)\right)\right|?$$



On complex exponential sum estimation
That's certainly not true as written. The right side is just
$$
\left | \left( \sum\_{2\le n\le N} e(cn)\log n \right)^2 \right|
= \left|\sum\_{2\le n\le N} e(cn)\log n \right|^2.
$$


If you choose $c$ to be 1/2, this is just the square of an alternating series (and hence is
bounded by $(\log N)^2$. 


On the other hand, if you choose $k=N$ on the left (say $N$ is even), you've got approximately $N(\log N)^2$.


This can be done with irrational $c$ as well if you take a bit of care.


