GAP Manual: 10.24. MoebiusMu

MoebiusMu( n )

MoebiusMu computes the value of the Moebius function for the integer n. This is 0 for integers which are not squarefree, i.e., which are divisible by a square r^2. Otherwise it is 1 if n has an even number and -1 if n has an odd number of prime factors.

The importance of mu stems from the so called inversion formula. Suppose f(n) is a function defined on the positive integers and let g(n)=sum_(d midn)(f(d)). Then f(n)=sum_(d midn)(mu(d) g(n/d)). As a special case we have phi(n) = sum_(d midn)(mu(d) n/d) since n = sum_(d midn)(phi(d)) (see Phi).

MoebiusMu usually spends all of its time factoring n (see FactorsInt). 

    gap> MoebiusMu( 60 );
    0
    gap> MoebiusMu( 61 );
    -1
    gap> MoebiusMu( 62 );
    1
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