The Mangoldt Function
Mangoldt's function is defined as
A short list of values of
is given below
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 0 |
|
|
|
| 0 |
|
|
| 0 |
The Mangoldt function arises naturally in the fundamental theorem of arithmetic. If we take natural logs of the prime power factorisation of
we obtain
The only nonzero terms in the sum come from those divisors
of the form
for
and
Hence
is a very useful formula.
We can invert this expression to obtain
in terms of natural logs using the following theorem.
Theorem
If
then
Proof:
where
The inverse to
is
the Mobius function so
for all
so
