I have been facing some difficulties with the following question.
For an absolutely convergent series $\sum_m a_m$, and the Möbius function $\mu(n)$, $x=(x_1,x_2)\in \mathbb{R}^2$, and $\alpha ,c\in \mathbb{R}$, if we have the following double sum $$\mathcal{S}=\sum_{m\in \mathbb{Z}^2}a_m\sum_{n\le N}\mu(n)e(m_1(x_1+n\alpha)+m_2(ncx_1+x_2)),$$ assuming that the inside is $o_{m_1,~m_2}(N)$, which means the constant may depend on $m_1,m_2$. Do we still have $\mathcal{S}=o(N)?$ How do we separate the $m$-terms from the inside?
Any comment is welcome! Thanks in advance!