The derivative of an arithmetical function is defined as the arithmetical function
for
Example
for all
Since
we can write
The definition of derivative given above obeys many of the rules obeyed by the ordinary derivative in calculus.
Theorem
a)
b)
c)
a)True by linearity of the summation.
b)
c)0=
so
Multiply both sides by
to give
since
Example
Differentiating
gives
Differentiating again gives
and using
gives
now multiply both sides by u^{-1}=%mu to give
This last equation is called the Selberg Identity.