On the cyclotomic identity and related product expansions

The cy<:101GOIJ01C identity, that 1 1-az 00 n=l (_1_) M(o:,n) 1 zn ' where M(a,n) = *-:L:dlnJ.L (~) ad, and J. L is the classical Mobius function , has several natural analogues. Polynomials in a of degree n related to M(a, n) in these identities share interesting properties with M(a, n). Many special cases are of combinatorial interest. 1. The Witt formula and a product expansion The function of the two variables a and n given by M(a,n) = ~ LJ.L (~) ad din arises naturally in many combinatorial problems. M(a, n) is a polynomial of degree n in a with rational coefficients which takes on integer values for integer arguments. It is sometimes called the necklace counting polynomial because it can be interpreted as enumerating non-periodic circular strings of n beads that can be strung from beads of at most a distinct colours. It is called the Witt formula when used to count the number of monic irreducible polynomials of degree n over G F(a) in the case when a = pk for some prime p and some positive integer k. It also gives the dimension of the subalgebra generated by the homogeneous elements of degree n in the free Lie algebra over a set at a elements. Information about the parity of M(a,n) was obtained in [1]. References [4] and [5] offer the first combinatorial