Möbius function

Definition

For any positive integer n, define μ(n) as the sum of the primitive nth roots of unity. It has values in {−1, 0, 1} depending on the factorization of n into prime factors:

μ(n) = −1 if n is a square-free positive integer with an even number of prime factors.

μ(n) = −1 if n is a square-free positive integer with an odd number of prime factors.

μ(n) = −0 if n has a squared prime factor.

The Möbius function can alternatively be represented as

μ ( n ) = δ ω ( n ) Ω ( n ) λ ( n )

where δ_ω(n)Ω(n) is the Kronecker delta, λ(n) is the Liouville function, ω(n) is the number of distinct prime divisors of n, and Ω(n) is the number of prime factors of n, counted with multiplicity.

The values of μ(n) for the first 30 positive numbers (sequence A008683 in the OEIS) are

The first 50 values of the function are plotted below:

Properties

The Möbius function is multiplicative (i.e. μ(ab) = μ(a) μ(b) whenever a and b are coprime). The sum of the Möbius function over all positive divisors of n (including n itself and 1) is zero except when n = 1:

∑ d | n μ ( d ) = { 1  if  n = 1 0  if  n > 1.

This is because the nth roots of unity sum to 0, and each nth root of unity is a primitive dth root of unity for exactly one divisor d of n.

The equality above leads to the important Möbius inversion formula and is the main reason why μ is of relevance in the theory of multiplicative and arithmetic functions.

Other applications of μ(n) in combinatorics are connected with the use of the Pólya enumeration theorem in combinatorial groups and combinatorial enumerations.

Mertens function

In number theory another arithmetic function closely related to the Möbius function is the Mertens function, defined by

M ( n ) = ∑ k = 1 n μ ( k )

for every natural number n. This function is closely linked with the positions of zeroes of the Riemann zeta function. See the article on the Mertens conjecture for more information about the connection between M(n) and the Riemann hypothesis.

There is a formula for calculating the Möbius function without directly knowing the factorization of its argument:

μ ( n ) = ∑ gcd ( k , n ) = 1 1 ≤ k ≤ n e 2 π i k n ,

i.e. μ(n) is the sum of the primitive n-th roots of unity. (However, the computational complexity of this definition is at least the same as that of the Euler product definition.)

From this it follows that the Mertens function is given by:

M ( n ) = ∑ a ∈ F n e 2 π i a ,

where F_n is the Farey sequence of order n.

This formula is used in the proof of the Franel–Landau theorem.

Proof of the formula for ∑d|n μ(d)

The formula given above,

∑ d | n μ ( d ) = { 1  if  n = 1 0  if  n > 1.

is trivially true when n = 1. Suppose then that n > 1. Then there is a bijection between the factors d of n for which μ(d) ≠ 0 and the subsets of the set of all prime factors of n. The asserted result follows from the fact that every non-empty finite set has an equal number of odd- and even-cardinality subsets.

This last fact can be shown easily by induction on the cardinality |S| of a non-empty finite set S. First, if |S| = 1, there is exactly one odd-cardinality subset of S, namely S itself, and exactly one even-cardinality subset, namely ∅. Next, if | S | > 1, then divide the subsets of S into two subclasses depending on whether they contain or not some fixed element x in S. There is an obvious bijection between these two subclasses, pairing those subsets that have the same complement relative to the subset {x}. Also, one of these two subclasses consists of all the subsets of the set S {x}, and therefore, by the induction hypothesis, has an equal number of odd- and even-cardinality subsets. These subsets in turn correspond bijectively to the even- and odd-cardinality {x}-containing subsets of S. The inductive step follows directly from these two bijections.

A related result is that the binomial coefficients exhibit alternating entries of odd and even power which sum symmetrically.

Mathematical series

The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function; if s is a complex number with real part larger than 1 we have

∑ n = 1 ∞ μ ( n ) n s = 1 ζ ( s ) .

This may be seen from its Euler product

1 ζ ( s ) = ∏ p ∈ P ( 1 − 1 p s ) = ( 1 − 1 2 s ) ( 1 − 1 3 s ) ( 1 − 1 5 s ) ⋯

When s is a complex number with real part larger than 1, the Dirichlet series for the Möbius function also satisfies:

∑ n = 1 ∞ μ ( n ) n s = 1 − ∑ a = 2 ∞ 1 a s + ∑ a = 2 ∞ ∑ b = 2 ∞ 1 ( a b ) s − ∑ a = 2 ∞ ∑ b = 2 ∞ ∑ c = 2 ∞ 1 ( a b c ) s + ∑ a = 2 ∞ ∑ b = 2 ∞ ∑ c = 2 ∞ ∑ d = 2 ∞ 1 ( a b c d ) s − ⋯

The Lambert series for the Möbius function is:

∑ n = 1 ∞ μ ( n ) q n 1 − q n = q ,

which converges for | q | < 1.

The ordinary generating function for the Möbius function follows from the binomial series

( I + X ) − 1

applied to triangular matrices:

∑ n = 1 ∞ μ ( n ) x n = x − ∑ a = 2 ∞ x a + ∑ a = 2 ∞ ∑ b = 2 ∞ x a b − ∑ a = 2 ∞ ∑ b = 2 ∞ ∑ c = 2 ∞ x a b c + ∑ a = 2 ∞ ∑ b = 2 ∞ ∑ c = 2 ∞ ∑ d = 2 ∞ x a b c d − ⋯

Algebraic number theory

Gauss proved that for a prime number p the sum of its primitive roots is congruent to μ(p − 1) (mod p).

If F_q denotes the finite field of order q (where q is necessarily a prime power), then the number N of monic irreducible polynomials of degree n over F_q is given by:

N ( q , n ) = 1 n ∑ d | n μ ( d ) q n d .

Average order

The average order of the Möbius function is zero. This statement is, in fact, equivalent to the prime number theorem.

μ(n) sections

μ(n) = 0 if and only if n is divisible by the square of a prime. The first numbers with this property are (sequence A013929 in the OEIS):

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63,....

If n is prime, then μ(n) = −1, but the converse is not true. The first non prime n for which μ(n) = −1 is 30 = 2 × 3 × 5. The first such numbers with three distinct prime factors (sphenic numbers) are:

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, … (sequence A007304 in the OEIS).

and the first such numbers with 5 distinct prime factors are:

2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 7854, 8610, 8778, 8970, 9030, 9282, 9570, 9690, … (sequence A046387 in the OEIS).

Incidence algebras

In combinatorics, every locally finite partially ordered set (poset) is assigned an incidence algebra. One distinguished member of this algebra is that poset's "Möbius function". The classical Möbius function treated in this article is essentially equal to the Möbius function of the set of all positive integers partially ordered by divisibility. See the article on incidence algebras for the precise definition and several examples of these general Möbius functions.

Popovici's function

Popovici defined a generalised Möbius function μ_k = μ ∗ μ ∗ μ to be the k-fold Dirichlet convolution of the Möbius function with itself. It is thus again a multiplicative function with

μ k ( p a ) = ( − 1 ) a ( k a )  

where the binomial coefficient is taken to be zero if a > k. The definition may be extended to complex k by reading the binomial as a polynomial in k.

Physics

The Möbius function also arises in the primon gas or free Riemann gas model of supersymmetry. In this theory, the fundamental particles or "primons" have energies log p. Under second quantization, multiparticle excitations are considered; these are given by log n for any natural number n. This follows from the fact that the factorization of the natural numbers into primes is unique.

In the free Riemann gas, any natural number can occur, if the primons are taken as bosons. If they are taken as fermions, then the Pauli exclusion principle excludes squares. The operator (−1)^F that distinguishes fermions and bosons is then none other than the Möbius function μ(n).

The free Riemann gas has a number of other interesting connections to number theory, including the fact that the partition function is the Riemann zeta function. This idea underlies Connes's attempted proof of the Riemann hypothesis.