Möbius function

The Möbius function (also known as Moebius μ - function ) is an important multiplicative function in number theory and combinatorics. It is named after the German mathematician August Ferdinand Möbius, who first introduced it in 1831. This function is a special case of a more general object of combinatorics.

Definition

The value is defined for all natural numbers and takes values ​​from the set. The function values ​​depend on the prime factorization of. Möbius function is defined as follows:

The function value is undefined in general.

Note: A number referred to as square-free when it has no divider is the square of a natural number greater than 1. This is equivalent to saying that every prime factor only appears exactly once.

Properties

• The Möbius function is the inverse function for the one- element with respect to the Dirichlet convolution.
• For all primes μ (n) = -1.
• For all square numbers μ ( n ) = 0
• μ (s) is multiplicative, i.e. μ ( a * b ) = μ ( a) · μ ( b) a and b are coprime
• For the summatorische function of the Möbius function applies to:

Wherein the sum runs over all divisor of n. It follows from the Möbius inversion formula.

Examples and values

• μ (7) = -1, since 7 is a prime number.
• μ (66 ) = (-1 ) 3 = -1 since 66 = 2 · 3 · 11
• μ (18 ) = 0, since 18 = 2 · 32 is not square-free.

The first 20 values ​​of the μ - function are ( sequence A008683 in OEIS ):

Image of the first 50 values ​​of the Möbius function:

Mertens function

The by Franz Mertens named Mertens function M ( n) represents a summation of the Möbius function μ (k ) is:

This corresponds to the difference of square-free numbers with an even number of prime factors to those with an odd number of prime factors of numbers up to n, the Mertens function ozilliert seemingly chaotic.

Zero crossings of the Mertens function can be found at: