let $x\in\mathbb{R}-\mathbb{Z}$ and $e(x)=e^{2\pi ix}$. If we have this sum $$\left|\overset{q}{\underset{h=1}{\sum}^{\*}}e\left(h\, x\right)\underset{\underset{p\equiv h\,\textrm{mod}\, q}{p\leq N}}{\sum}\log p\right|$$where ${\sum}^{\*}$ is the sum with the condition $\left(h,q\right)=1$ can we affirm that exist a $h^{\*}\in\left[1,q\right],\,\left(h^{\*},q\right)=1$ such that $$\left|\overset{q}{\underset{h=1}{\sum}^{\*}}e\left(h\, x\right)\underset{\underset{p\equiv h\,\textrm{mod}\, q}{p\leq N}}{\sum}\log p\right|\leq\left|\overset{q}{\underset{h=1}{\sum}^{\*}}e\left(h\, x\right)\right|\underset{\underset{p\equiv h^{\*}\,\textrm{mod}\, q}{p\leq N}}{\sum}\log p$$ so there exists a $h^{\*}$ that maximizes this sum? Thank you.



trigonometric sum and inequalities
No. If $q$ is not square-free and $x=1/q$, then the right hand side of your inequality is zero, while the left hand side is surely not.


