In mathematics, the cyclotomic identity states that
1 1 − k z = ∏ j = 1 ∞ ( 1 1 − z j ) M j ( k ) {\displaystyle {1 \over 1-kz}=\prod _{j=1}^{\infty }\left({1 \over 1-z^{j}}\right)^{M_{j}(k)}}
where M is Moreau's necklace-counting function,
M n ( k ) = 1 n ∑ d | n μ ( n d ) k d , {\displaystyle M_{n}(k)={1 \over n}\sum _{d\,|\,n}\mu \!\left({n \over d}\right)k^{d},}
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 k generators, and the right hand side is the generating function for the universal enveloping algebra of the free Lie algebra on k generators. The cyclotomic identity witnesses the fact that these two algebras are isomorphic. Another interpretation is as the Hasse–Weil zeta function of the affine line over the finite field F q {\displaystyle \mathbb {F} _{q}} with k = q {\displaystyle k=q} elements. The exponent M n ( k ) {\displaystyle M_{n}(k)} counts the number of number of maximal ideals of degree n in the coordinate ring F q [ x ] {\displaystyle \mathbb {F} _{q}[x]} , i.e. the number of monic irreducible polynomials of degree n. There is also a symmetric generalization of the cyclotomic identity found by Strehl:
∏ j = 1 ∞ ( 1 1 − k z j ) M j ( ℓ ) = ∏ j = 1 ∞ ( 1 1 − ℓ z j ) M j ( k ) {\displaystyle \prod _{j=1}^{\infty }\left({1 \over 1-kz^{j}}\right)^{M_{j}(\ell )}=\prod _{j=1}^{\infty }\left({1 \over 1-\ell z^{j}}\right)^{M_{j}(k)}}
References 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