gives the Möbius function .
- MoebiusMu is also known as Möbius function.
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- MoebiusMu[n] gives the sum of the primitive roots of unity.
- For a number n=u p1k1⋯ pmkm with u a unit and pi primes, MoebiusMu[n]returns 0 unless all ki are equal to 1, in which case it gives (-1)m.
- MoebiusMu[m+In] automatically works over Gaussian integers.
Examplesopen allclose all
Basic Examples (2)
Compute the Möbius function at 11:
Plot the MoebiusMu sequence for the first 20 numbers:
Numerical Manipulation (4)
Symbolic Manipulation (6)
Use MoebiusMu in a sum:
Identify the MoebiusMu sequence:
DirichletTransform of MoebiusMu:
Basic Applications (2)
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:
Number Theory (9)
Use MoebiusMu to test for a square-free number:
Use MoebiusMu to compute the number of terms in the Farey sequence:
Use MoebiusMu to compute MangoldtLambda:
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]:
Properties & Relations (7)
MoebiusMu is a multiplicative function:
is 1 if n is a product of an even number of distinct primes:
is if it is a product of an odd number of primes:
is 0 if it has a multiple prime factor:
MoebiusMu is 0 for composite prime powers and for primes:
MoebiusMu is 0 for non-square-free integers:
Use PrimeNu to compute MoebiusMu for square-free numbers:
MoebiusMu is equal to the sum of the primitive roots of unity:
MoebiusMu can be expressed in terms of LiouvilleLambda and KroneckerDelta:
Wolfram Research (1988), MoebiusMu, Wolfram Language function, https://reference.wolfram.com/language/ref/MoebiusMu.html.
Wolfram Language. 1988. "MoebiusMu." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MoebiusMu.html.
Wolfram Language. (1988). MoebiusMu. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MoebiusMu.html