example, consider the case . Then
:

By Möbius inversion, we have
:where is the Möbius function. Then by using the product rule for the logarithm we get
:

For all , we have
:Also, there exist positive constants and such that
:for all , and
:for all sufficiently large .

Dirichlet series

The von Mangoldt function plays an important role in the theory of Dirichlet series,