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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?

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

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