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In mathematics, the **cyclotomic identity** states that

where *M* is Moreau's necklace-counting function,

and *μ* is the classic Möbius function of number theory.

The name comes from the denominator, 1 − *z*^{ j}, which is the product of cyclotomic polynomials.

The left hand side of the cyclotomic identity is the generating function for the free associative algebra on α generators, and the right hand side is the generating function for the universal enveloping algebra of the free Lie algebra on α generators. The cyclotomic identity witnesses the fact that these two algebras are isomorphic.

There is also a symmetric generalization of the cyclotomic identity found by Strehl:

- Metropolis, N.; Rota, Gian-Carlo (1984), "The cyclotomic identity", in Greene, Curtis (ed.),
*Combinatorics and algebra (Boulder, Colo., 1983). Proceedings of the AMS-IMS-SIAM joint summer research conference held at the University of Colorado, Boulder, Colo., June 5–11, 1983.*, Contemp. Math., vol. 34, Providence, R.I.: American Mathematical Society, pp. 19–27, ISBN 978-0-8218-5029-9, MR 0777692