where the sum is over all positive divisors of n and [mu] is the Mobius function
, which is an arithmetic function defined by (cf.
Recall the one-variable Mobius function
of a poset, [mu]: P [right arrow] Z, is defined recursively by
This product has a unit, and the constant sequence [zeta] = (1, 1, ...) has a convolution inverse: the classical Mobius function
[mu], given by a well-known formula involving prime factorizations.
The e-convolution [??] is commutative, associative and has the identity element [[mu].sup.2], where [mu] is the Mobius function
. The inverse with respect to [??] of the constant 1 function is called the exponential analogue of the Mobius function
and it is denoted by [[mu].sup.(e)].
Obviously, [[mu].sub.1](n) = [mu](n), n [member of] N, is the classical Mobius function
: = 1; if n is not square free then [mu](n) = 0; if n is square free and if q is the number of distinct primes dividing n, then [mu](n) = [(-1).sup.q].
He analyzes the relevance of algebraic structures to number theory in such topics as ordered fields, fields with valuation and other algebraic structures, the role of the Mobius function
and of generating functions, semigroups and certain convolution algebras.
(Recall that if C = ([c.sub.ij]) is an m by r matrix and D is a p by q matrix, then die tensor product of C and D is the mp by rq matrix consisting of an m by r block array of p by q blocks where the ij block is [c.sub.ij] D.) The inverse of the zeta function is called the mobius function
of the power set of S.
A classical question about any combinatorially defined poset is what its Mobius function
[mu](n) denotes the Mobius function
. For fixed integers 1 [less than or equal to] a [less than or equal to] b, the divisor function d(a, b; n) is defined by
Second, the subset expansion is a summation of Mobius function
values over the lattice of graphs, ordered by inclusion of edge sets.
where [mu](n) is the Mobius function
, [d.sub.1] and [d.sub.2] are computable constants.
Indeed, when the increasing flip graph is the Hasse diagram of the increasing flip poset, this poset is EL-shellable, and we can compute its Mobius function
. These results extend recent work of M.