Definition (Arithmetic function)

An arithmetic function is defined as $f:\mathbb{N}\to A\subset\mathbb{C}$ or as a function mapping the positive integers to some subset of the complex numbers.

Definition (Multiplicative)

An arithmetic function, $f$, is said to be multiplicative if $$f(ab)=f(a)f(b)$$for all $a,b\in\mathbb{N}$ coprime (i.e. $(a,b)=1$).

Definition (Additive)

An arithmetic function is said to be additive if $$f(ab)=f(a)+f(b)$$

Definition (Binary function)

The arithmetic function $$f_{b}(n)=\left[ \frac{\log(n)}{\log(b)} \right]+1$$where, $[x]$ is the floor function. This function takes a natural number and tells us the number of digits of $n$ in its expansion in base $b$.

Definition (Positiver divisor counter function)

The arithmetic function $d(n)$ counts the number of positive divisors of $n$ and is defined as $$d(n)=\prod_{i=1}^{k}(\alpha_{i}+1)$$where we recall by the Fundamental Theorem of Arithmetic that $$n=\prod_{i=1}^{k}p_{i}^{\alpha_{i}}$$

Definition (Euler’s totient function)

Euler’s function, an arithmetic function, $\phi(n)$ finds the number of positive numbers less than $n$ and coprime to $n$ this function is defined as follows: $$\frac{\phi(n)}{n}=\prod_{p|n}\left( 1-\frac{1}{p} \right)$$where $p$ are the distinct prime numbers dividing $n$. Consequently, the function is Multiplicative.