sábado, 1 de noviembre de 2014

FUNCIONES ARITMÉTICAS

FUNCIÓN : DE MÖBIUS .- ......................:http://es.wikipedia.org/w/index.php?title=Funci%C3%B3n_de_M%C3%B6bius&printable=yes

Möbius Function

The Möbius function is a number theoretic function defined by

$mu(n)={0 if n has one or more repeated prime factors; 1 if n=1; (-1)^k if n is a product of k distinct primes,$

(1)

indicates that

is squarefree (Havil 2003, p. 208). The first few values of

are therefore 1,

, 0,

, 1,

, 0, 0, 1,

, 0, ... (OEIS A008683). Similarly, the first few values of

for

, 2, ... are 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, ... (OEIS A008966).

The function was introduced by Möbius (1832), and the notation

was first used by Mertens (1874). However, Gauss considered the Möbius function more than 30 years before Möbius, writing "The sum of all primitive roots [of a prime number

] is either

(when

is divisible by a square), or

(mod

) (when

is the product of unequal prime numbers; if the number of these is even the sign is positive but if the number is odd, the sign is negative)" (Gauss 1801, Pegg 2003).

The Möbius function is implemented in Mathematica as MoebiusMu[n].

The summatory function of the Möbius function

(2)

is called the Mertens function.

The following table gives the first few values of

for

, 0, and 1. The values of the first

integers are plotted above on a

grid, where values of

with

are shown in red,

are shown in black, and

are shown in blue. Clear patterns emerge where multiples of numbers each share one or more repeated factor.

	Sloane	values of
	A030059	2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, ...
0	A013929	4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, ...
1	A030229	1, 6, 10, 14, 15, 21, 22, 26, ...

The Möbius function has generating functions

(3)

for

(Nagell 1951, p. 130). This product follows by taking one over the Euler product and expanding the terms to obtain

			(4)
			(5)
			(6)
			(7)
			(8)

(Derbyshire 2004, pp. 245-249).

An additional generating function is given by

(9)

for

. It also obeys the infinite sums

			(10)
			(11)
			(12)
$sum_(n=1)^(infty)(chi_({n:mu(n)=-1}))/(n^2)$			(13)
$sum_(n=1)^(infty)(chi_({n:mu(n)=1}))/(n^2)$			(14)

(OEIS A082020, A088245, and A088245; Havil 2003, p. 208), as well as the divisor sum

(15)

where

is the number of distinct prime factors of

(Hardy and Wright 1979, p. 235).

also satisfies the infinite product

(16)

for

(Bellman 1943; Buck 1944;, Pólya and Szegö 1976, p. 126; Robbins 1999). Equation (◇) is as "deep" as the prime number theorem (Landau 1909, pp. 567-574; Landau 1911; Hardy 1999, p. 24).

The Möbius function is multiplicative,