## Derivatives of Arithmetic Functions

The derivative of an arithmetical function is defined as the arithmetical functionfor

Example

for all

Sincewe 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 byto givesince

Example

DifferentiatinggivesDifferentiating again givesand usinggivesnow multiply both sides by u^{-1}=%mu to give

This last equation is called the Selberg Identity.