# Bounds for a character sum $\sum_{n } \dfrac{\eta(n/N)\chi(n)\sin(2\pi \delta n)}{n}$

Let $$\chi$$ be a primitive Dirichlet character of large modulus $$q > 1$$, and let $$\delta \in \mathbb{R}$$ be fixed. Assume $$\eta : \mathbb{R} \to [0,1]$$ is smooth compactly supported on $$[1,2]$$. For a large real number $$N \geq 1$$, consider the sum $$S_N(\delta,\chi) = \sum_{n =1}^\infty \dfrac{\eta(n/N)}{n}\chi(n)\sin(2\pi \delta n).$$ I'm wondering what are good upper bounds for the absolute value of $$S_N(\delta,\chi)$$. The trivial bound gives $$S_N(\delta,\chi) \ll 1$$ and perhaps one could even show that $$S_N(\delta,\chi) \ll 1/N$$. But can we do better?

• What for ? Do you know a method that can be adapted ? What do you get from the inverse Mellin transform of $L(s+1,\chi) \int_0^\infty \eta(x) \sin(2\pi \delta/x) x^{-s-1}dx$ – reuns Mar 14 at 18:43