Examples of multiplicative functions include many important functions in number theory, such as:

• $\varphi(n)$: the Euler totient function (also denoted $\phi(n)$), counting the totatives of $n$;

• $\mu(n)$: the Möbius function, which determines the parity of the prime factors of $n$ if $n$ is squarefree;

• $\tau(n)$: the divisor function (also denoted $d(n)$), counting the positive divisors of $n$;

• $\sigma(n)$: the sum of divisors function (also denoted $\sigma_{1}(n)$), summing the positive divisors of $n$;

• $\sigma_{{k}}(n)$: the sum of the $k$-th powers of all the positive divisors of $n$ for any complex number $k$ (typically a natural number);

• $\hbox{id}(n)$: the identity function, defined by $\hbox{id}(n)=n$;

• $\hbox{id}^{{k}}(n)$: the power functions, defined by $\hbox{id}^{{k}}(n)=n^{{k}}$ for any complex number $k$ (typically a natural number);

• $1(n)$: the constant function, defined by $1(n)=1$;

• $\varepsilon(n)$: the convolution identity function, defined by:

  $$\varepsilon(n)=\sum_{{d|n}}\mu(d)=\begin{cases}1&\text{if }n=1\\ 0&\text{if }n\neq 1\end{cases}$$

  where $d$ runs through the positive divisors of $n$.

A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic. Thus, if $n$ is a product of powers of distinct prime numbers, say $n=p^{a}q^{b}\cdots$, then $f(n)=f(p^{a})f(q^{b})\cdots$. This property of multiplicative functions significantly reduces the need for computation, as in the following examples for $n=144=2^{4}\cdot 3^{2}$:

Similarly, we have:

Recall that, if $f$ and $g$ are two arithmetic functions, one defines a new arithmetic function $f*g$, the Dirichlet convolution (or simply convolution) of $f$ and $g$, by

where the sum extends over all positive divisors $d$ of $n$. Some general properties of this operation with respect to multiplicative functions include (here the argument $n$ is omitted in all functions):

• If both $f$ and $g$ are multiplicative, then so is $f*g$ (proven here);

• $f*g=g*f$ (proven here);

• $(f*g)*h=f*(g*h)$ (proven here);

• $f*\varepsilon=\varepsilon*f=f$ (proven here);

• If $f$ is multiplicative, there exists a multiplicative function $g$ such that $f*g=\varepsilon$ (proven here). In other words, every multiplicative function has a convolution inverse that is also multiplicative.

This shows that, with respect to convolution, the multiplicative functions form an abelian group with identity element $\varepsilon$. Relations among the multiplicative functions discussed above include:

• $\mu*1=\varepsilon$ (the Möbius inversion formula)

• $1*1=\tau$

• $\hbox{id}*1=\sigma$

• $\hbox{id}^{{k}}*1=\sigma^{{k}}$

• $\phi*1=\hbox{id}$

Given a completely multiplicative function $f$, its convolution inverse is $f\mu$. See this entry for a proof.