The Mobius function is one of a class of arithmetic functions – complex or real valued functions defined on the positive integers.
The Mobius function is labelled
and defined as
If
then write
Then
otherwise.
Note that
if and only if
has a square factor greater than 1.
Values of
for
to 10 are given in the following table.
|
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
|
1 |
-1 |
-1 |
0 |
-1 |
1 |
-1 |
0 |
0 |
1 |
The most fundamental property of the Mobius function is that it gives a simple formula for the divisor sum
extended over the positive divisors of
Theorem
If
then
Proof: The formula is clearly true ifSuppose then that
In the sumthe only nonzero terms come from
and those divisors ofwhich are products of distinct primes. Thus
The Mobius function is not multiplicative, so does not obey the formula
To see this put
then
but