Equivalence of Büchi automata and linear $\mu$-calculus
The constructive equivalence of linear-time fixed point formulae (the logic is called $\nu$TL by some) and Buechi automata is given in a paper by Mads Dam from 1992.

Fixed Points of Buchi Automata, FST&amp;TCS 1992.

See page 4 for the construction of a $\nu$TL formula from a Buechi automaton. The construction of a Buechi automaton from a $\nu$TL formula is more complicated and takes the rest of the paper.
The rest of this reply is a brief argument that this result existed in the literature in far less direct form. Pierre Wolper showed that there were omega-regular properties that were not LTL-definable and gave an extension of LTL (called ETL) that could express omega-regular properties.

Temporal Logic can be more expressive, Pierre Wolper, Information and Computation, 1983.

It is also known that one can translate ETL formulae into $\nu$TL formulae, so by combining these results you can read off a translation of Buechi automata into $\nu$TL. In the other direction, it follows from the work of Buechi that S1S (the second order theory of one successor) formulae can be compiled into Buechi automata and by translating $\nu$TL formulae into S1S, we obtain a translation of $\nu$TL to Buechi automata.
If you want a more in-depth introduction to these topics, I suggest Mads Dam's lecture notes, or the work of Roope Kaivola (sadly not as widely known as much related work).

Temporal Logics, Automata, and Classical Theories - An Introduction, Mads Dam, ESSLLI 1994.
Using Automata to Characterise Fixed Point Temporal Logics, Roope Kaivola