Arithmetic Functions

Def Arithmetic Function
An arithmetic function is generally any function $f:{\mathbb{Z}}^{+}\to \text{C}$.

Def Additivity
An arithmetic function $f$ is completely additive if $f(mn)=f(m)+f(n)$ for all natural numbers $m$ and $n$; It is additive if $f(mn)=f(m)+f(n)$ for all coprime natural numbers $m$ and $n$.

Def Multiplicativity
An arithmetic function $f$ is completely multiplicative if $f(mn)=f(m)f(n)$ for all natural numbers $m$ and $n$; It is multiplicative if $f(mn)=f(m)f(n)$ for all coprime natural numbers $m$ and $n$.

Euler Totient Function

The Euler totient function $\phi (q):{\mathbb{Z}}^{+}\to \mathbb{N}$ is mapping to the number of positive integers not greater than $q$ that are coprime to $q$: $\phi (q)=n{\prod }_{p\mid q}\left(1-\frac{1}{p}\right)$

Prop The Euler totient function is multiplicative.

Def Dirichlet Character
A complex-valued arithmetic function $\chi :\mathbb{Z}\to \mathbb{C}$ is a Dirichlet character of modulus $q$ if for all integers $a$ and $b$:

• $\chi (ab)=\chi (a)\chi (b)$ i.e. $\chi$ is completely multiplicative.
• $\chi (a)=0$ for all $\gcd (a,q)>1$
• $\chi (a)\ne 0$ for all $\gcd (a,q)=1$
• $\chi (a+q)=\chi (a)$
  We call trivial character the principle character, denoted as ${\chi }_0(a)=\{\begin{array}{l}0\gcd (a,q)>1\\ 1\text{otherwise}\end{array}$