See the second and third paragraphs after the identity labeled as $(10)$ of this What's new: The Euler-Maclaurin formula ..., by Terence Tao.
Question. Let then $\eta(x)$ a cutoff function, and $\mu(n)$ the Möbius function. Is it possible calculate the asymptotic behaviour of $$\sum_{n=1}^\infty\eta\left(\frac{n}{N}\right)\frac{\mu(n)}{n^2}$$ as the integer $N\to\infty$?
(And if additionally you need to assume that our cutoff function satisfies more conditions to get a more precise asymptotic identity feel free to argue such calculations and conditions). Many thanks.
Please if such calculation is feasible provide hints or details, as you want. Feel free to add clarifications of your words and details of the theorems that you use.