Farey Sequence and Mertens function

Question

Mertens function $M(n)$ is defined as the cumulative sum of Möbius functions $\mu(k)$: $$M(n)=\sum_{k=1}^n\mu(k)$$ and is profoundly related to the Riemann hypothesis. A nice alternative formula (though not very usefull computational wise) uses the Farey sequence $F_n$ of completely reduced fractions to write $$M(n)=-1+\sum_{k\in F_n}\mathrm{e}^{2\pi\mathrm{i}k}.$$ During a work on Fermi surfaces I came across the following sum in two dimensions: $$\sum_{\vec{k}\in F_n\times F_n}\!\!\!\!\mathrm{e}^{2\pi\mathrm{i}\vec{k}\cdot\vec{x}}=\biggl(\sum_{k_x\in F_n}\mathrm{e}^{2\pi\mathrm{i}k_xx}\biggr)\biggl(\sum_{k_y\in F_n}\mathrm{e}^{2\pi\mathrm{i}k_yy}\biggr).$$ I was wondering if someone knows a relation (if there is one...) between my goal sum and Mertens function, i.e. generalizations of the Farey sum definition of $M(n)$?