## The Mangoldt Function

Mangoldt's function is defined as

A short list of values ofis 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 ofwe obtain

The only nonzero terms in the sum come from those divisorsof the formfor and

Hence

is a very useful formula.

We can invert this expression to obtainin terms of natural logs using the following theorem.

Theorem

Ifthen

Proof:whereThe inverse toisthe Mobius function so

for allso