The Möbius transform maps a sequence 
, 
, ... satisfying
 |
(1)
|
for positive integers 
to the sequence 
, 
, ... via the Möbius
inversion formula,
 |
(2)
|
where 
is the Möbius function. The transformation
of the sequence 
to 
is sometimes called the sum-of-divisors transform. An equivalent
generating function identity is
 |
(3)
|
the right side of which is called a Lambert series. This is an identity of formal power series. The corresponding Dirichlet generating function identity is
 |
(4)
|
where 
is the Riemann zeta function. The analytic
identity holds when ![Re[s]>1](http://mathworld.wolfram.com/images/equations/MoebiusTransform/Inline10.svg)
and the series 
converges absolutely
(Sloane and Plouffe 1995, p. 21).
Example Möbius transformations (Sloane and Plouffe 1995, p. 22) include 
for all 
,
giving the inverse transform as 
, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, ... (OEIS A000005),
the divisor function 
of 
. The Möbius transform of 
gives 
, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, ... (OEIS A000010),
the totient function of 
. The inverse Möbius transform of the sequence 
for 
and 
for 
gives 
, 4, 0, 4, 8, 0, 0, 4, 4, ... (OEIS A004018),
the sum of squares function 
counting signed, ordered representations of 
as a sum of two squares.
The inverse Möbius transform of 
for prime 
and 
otherwise gives the sequence 
, 1, 1, 1, 1, 2, 1, 1, 1, ... (OEIS
A001221), the number of distinct
prime factors of 
.
