MoebiusMu works over integers:

Gaussian integers:

Compute for large integers:

MoebiusMu threads over lists:

TraditionalForm formatting:

Reduce expressions:

Solve equations:

Use MoebiusMu in a sum:

In a product:

Identify the MoebiusMu sequence:

DirichletTransform of MoebiusMu:

Equivalently:

Highlight numbers n for which in blue, numbers n for which in red and numbers n for which in black:

Histogram of the cumulative values of MoebiusMu:

The underlying distribution from the data:

Use MoebiusMu to test for a square-free number:

Use MoebiusMu to compute the number of terms in the Farey sequence:

Compare with the following:

Use MoebiusMu to compute MangoldtLambda:

LiouvilleLambda:

EulerPhi:

KroneckerDelta:

gives 1 when and 0 when :

MoebiusMu is related to DivisorSigma through the Möbius inversion formulas:

MoebiusMu is related to PrimeNu through the following formula:

MoebiusMu satisfies the following identities:

Compute the number of polynomials over that are irreducible of degree n:

Irreducible polynomials modulo 5:

Distribution of irreducible polynomials modulo 5:

Logarithmic plot of the count for :

Plot the Mertens function [more info]:

Plot :