Möbius function

The Möbius function (also called Möbius 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 the more generally defined Möbius function of a partial order , whereby the partial order on which this is based results from divisibility relations.

definition

The value is defined for all natural numbers and takes values ​​from the set . The function values ​​depend on the prime factorization of . The Möbius function is defined as follows: ${\displaystyle \mu (n)}$ ${\displaystyle n}$ ${\displaystyle \{-1,0,1\}}$ ${\displaystyle n}$

${\displaystyle \mu (n)={\begin{cases}(-1)^{k},&{\mbox{wenn }}n{\mbox{ quadratfrei, wobei }}k{\mbox{ die Anzahl der Primfaktoren von }}n{\mbox{ ist}},\\0&{\mbox{sonst.}}\end{cases}}}$

The function value remains undefined or is set to. ${\displaystyle \mu (0)}$ ${\displaystyle 0}$

Note: A number is said to be square-free if it does not have a factor that is the square of a natural number greater than 1. This means that every prime factor occurs exactly once.

properties

• The Möbius function is the inverse element to the one function with regard to the Dirichlet convolution .
• The following applies to all prime numbers . ${\displaystyle \mu (n)=-1}$
• ${\displaystyle \mu }$is multiplicative , i.e. i.e., for and coprime. ${\displaystyle \mu (a\cdot b)=\mu (a)\cdot \mu (b)}$ ${\displaystyle a}$ ${\displaystyle b}$
• The following applies to the summation function of the Möbius function : ${\displaystyle n\geq 2}$

${\displaystyle \sum \limits _{d|n}\mu (d)=0}$,

where the sum runs over all divisors of . Möbius' inverse formula also follows from this . ${\displaystyle n}$

Examples and values

• ${\displaystyle \mu (7)=-1}$since is a prime number. ${\displaystyle 7}$
• ${\displaystyle \mu (66)=(-1)^{3}=-1}$, there . ${\displaystyle 66=2\cdot 3\cdot 11}$
• ${\displaystyle \mu (18)=0}$, because it is not square-free. ${\displaystyle 18=2\cdot 3^{2}}$

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

n	1	2	3	4th	5	6th	7th	8th	9	10	11	12	13	14th	15th	16	17th	18th	19th	20th
μ (n)	1	−1	−1	0	−1	1	−1	0	0	1	−1	0	−1	1	1	0	−1	0	−1	0

${\displaystyle \mu (n)=-1}$	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, ...	(Follow A030059 in OEIS )
${\displaystyle \mu (n)=0}$	4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, ...	(Follow A013929 in OEIS )
${\displaystyle \mu (n)=1}$	1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, ...	(Follow A030229 in OEIS )

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

Mertens function

The Mertens function named after Franz Mertens represents a summation over the Möbius function: ${\displaystyle M}$

${\displaystyle M(n)=\sum _{k=1}^{n}\mu (k)}$

This corresponds to the difference between the number of square-free numbers with an even number of prime factors and the number of those with an odd number of prime factors up to the number . The Mertens function seems to oscillate chaotically. ${\displaystyle n}$

Zero crossings of the Mertens function can be found at:

2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, ... (sequence A028442 in OEIS ).

Assumptions about the asymptotic behavior of Möbius and Mertens functions are related to the Riemann Hypothesis , which is equivalent to the following statement:

${\displaystyle |M(x)|=\left|\sum _{n\leq x}\mu (n)\right|=O(x^{{\frac {1}{2}}+o(1)})}$

using the Landau symbols . The statement

${\displaystyle |M(x)|=\left|\sum _{n\leq x}\mu (n)\right|=o(x)}$

is equivalent to the prime number theorem according to Edmund Landau .

Chowla and Sarnak conjecture

The Chowla conjecture can be formulated for both the Liouville function and the Möbius function:

${\displaystyle \sum _{1\leq n\leq x}\mu (n+h_{1})^{a_{1}}\,\dotsb \,\mu (n+h_{k})^{a_{k}}=o(x)}$

for any natural numbers and , where not all are straight (where you look for on can limit). means asymptotically vanishing with (see Landau symbols ). If only one of the numbers is odd, that is equivalent to the prime number theorem in arithmetic progressions. Otherwise the assumption is open. ${\displaystyle h_{i}}$ ${\displaystyle a_{i}\geq 0}$ ${\displaystyle a_{i}}$ ${\displaystyle {\mu (n)}^{a+2}=\mu (n)^{a}}$ ${\displaystyle a_{i}=0,\,1,\,2}$ ${\displaystyle o(x)}$ ${\displaystyle x\to \infty }$ ${\displaystyle a_{i}}$

Another conjecture that describes the random behavior of the signs of the Möbius function is that of Peter Sarnak . Let be a complex-valued, bounded arithmetic function that is deterministic (the topological entropy of the sequence vanishes). Then according to the Sarnak conjecture: ${\displaystyle f(n)\in \mathbb {C} }$

${\displaystyle \sum _{n\leq x}f(n)\,\mu (n)=o(x)}$

It is generally open, but special cases are known. For a constant sequence this is essentially the prime number theorem , for periodic sequences the prime number theorem in arithmetic progressions, for quasi-periodic sequences this follows from a theorem by Harold Davenport and for horocycle flows from a theorem by Sarnak, Tamar Ziegler and Jean Bourgain . The Sarnak conjecture follows Sarnak from the Chowla conjecture.

Other uses

It plays a role in the fermion version of the toy model for the interpretation of the Riemann zeta function in primon gas .

Web links

• Eric W. Weisstein : Möbius Function . In: MathWorld (English).
• Eric W. Weisstein : Mertens Function . In: MathWorld (English).

literature

• Peter Bundschuh : Introduction to Number Theory. 6th edition. Springer, Berlin 2008, ISBN 978-3540764908 .

Individual evidence

1. ^ Terence Tao: The Chowla conjecture and the Sarnak conjecture. 2012 (Tao blog).
2. ↑ For definition, see Tao's blog cited above. If with for a compact metric space and a homeomorphism of , which together define a dynamic system, then this corresponds to the usual topological entropy of the dynamic system. ${\displaystyle f(n)=f(T^{n}(x))}$ ${\displaystyle x\in X}$ ${\displaystyle X}$ ${\displaystyle T}$ ${\displaystyle X}$