Cyclotomic identity

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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)}} {\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},} {\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}} {\displaystyle \mathbb {F} _{q}} with k = q {\displaystyle k=q} {\displaystyle k=q} elements. The exponent M n ( k ) {\displaystyle 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]} {\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)}} {\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)}}

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