I already know that $\mathbb{Q}/\mathbb{Z}$ has $\prod_{p\textrm{ prime}}\mathbb{Z}_{p}$ as its Pontryagin dual, and I'm pretty sure that $\mathbb{Q}/\mu\mathbb{Z}$ will depend on the prime divisors of $\mu$, but I can't figure it out, nor can I find an answer anywhere else on the internet. Formally, I get that the answer should be $\prod_{p\textrm{ prime}}\mu\mathbb{Z}\left(p^{\infty}\right)$, but I can't find any resource about what manipulations are valid when working with cosets of Prüfer $p$-groups. I'd like the answer, please, so that I can get on with my research. (Note: here, "answer" means answer, not "hint", or "patronizing remark about how I should be able to figure it out myself".)
Anyhow, thanks in advance.
$\mathbb{Q}/ \mathbb{Z}$ is isomorphic (as a topological group) to $\mathbb{Q}/\mathbb{\mu Z}$, the isomorphism is given by multiplication by $\mu$. Thus the pontrjagin dual to $\mathbb{Q}/\mathbb{\mu Z}$ is $(\lim\limits_{\rightarrow}\mathbb{Z}/\mathbb{nZ})\hat{} = \lim\limits_{\leftarrow}(\mathbb{Z}/\mathbb{nZ})\hat{} = \prod\limits_{p\textrm{ prime}}\mathbb{Z}_{p}$