Arithmetic Functions

Def Arithmetic Function An arithmetic function is generally any function .

Def Additivity An arithmetic function is completely additive if for all natural numbers and ; It is additive if for all coprime natural numbers and .

Def Multiplicativity An arithmetic function is completely multiplicative if for all natural numbers and ; It is multiplicative if for all coprime natural numbers and .

Euler Totient Function

The Euler totient function is mapping to the number of positive integers not greater than that are coprime to :

Prop The Euler totient function is multiplicative.

Def Dirichlet Character A complex-valued arithmetic function is a Dirichlet character of modulus if for all integers and :

• i.e. is completely multiplicative.
• for all
• for all
• We call trivial character the principle character, denoted as