The Möbius function is a number theoretic function defined by
 |
(1)
|
so 
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 the Wolfram Language 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.
 | OEIS | 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 ![R[s]>1](/images/equations/MoebiusFunction/Inline29.svg)
(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)
|
 |  |  |
(13)
|
 |  |  |
(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,
 |
(17)
|
and satisfies
 |
(18)
|
where 
is the Kronecker delta, as well as
 |
(19)
|
where 
is the number of divisors (i.e., divisor function
of order zero; Nagell 1951, p. 281).
