Möbius function

is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832.

[i][ii][2] It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula.

Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted

Another characterization by Gauss is the sum of all primitive roots.

is a complex number with real part larger than 1 we have This may be seen from its Euler product Also: The Lambert series for the Möbius function is which converges for

In this theory, the fundamental particles or "primons" have energies

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.

that distinguishes fermions and bosons is then none other than the Möbius function

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 Alain Connes's attempted proof of the Riemann hypothesis.

, so The sum of the Möbius function over all positive divisors of

: The equality above leads to the important Möbius inversion formula and is the main reason why

is of relevance in the theory of multiplicative and arithmetic functions.

There is a formula[7] for calculating the Möbius function without directly knowing the factorization of its argument: i.e.

Other identities satisfied by the Möbius function include and The first of these is a classical result while the second was published in 2020.

[8][9] Similar identities hold for the Mertens function.

The asserted result follows from the fact that every non-empty finite set has an equal number of odd- and even-cardinality subsets.

, 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

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

The mean value (in the sense of average orders) of the Möbius function is zero.

This statement is, in fact, equivalent to the prime number theorem.

[11] 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.

Constantin Popovici[12] defined a generalised Möbius function

It is thus again a multiplicative function with where the binomial coefficient is taken to be zero if