An arithmetic function is a complex-valued function defined on the set of natural numbers, i.e., it is of the form $f : \mathbb{N} \to \mathbb{C}$. In number theory, one is usually interested in those arithmetic functions $f$ for which $f(n)$ depends on some number-theoretic property of $n$. We list some arithmetic functions below.
- The identity function $e$ is the characteristic function of $1$, i.e., $e(1) = 1$ and $e(n) = 0$ for $n > 1$. We can write it as $e(n) = \lfloor 1/n \rfloor$.
- For $\alpha \in \mathbb{C}$ the power function $N^{\alpha}$ is defined as $N^{\alpha}(n) = n^{\alpha}$.
- The function $\omega(n)$ counts the number of distinct prime divisors of $n$ and $\Omega(n)$ counts the total number of prime divisors of $n$ (counted with multiplicity).
- The Liouville function $\lambda$ is defined as $\lambda(n) = (-1)^{\Omega(n)}$.
- For $\alpha \in \mathbb{C}$ the (generalized) divisor function $\sigma_{\alpha}$ is defined as $\sigma_{\alpha}(n) = \sum_{d \, | \, n} d^{\alpha}$.
- The radical function is defined as $\text{rad}(n) = \prod_{p \, | \, n} p $.
Given two arithmetic functions $f$ and $g$, the Dirichlet convolution $f*g$ is an arithmetic function defined as $$(f * g)(n) = \sum_{d \, | \, n} f(d) g(n/d).$$ It is easily verified that the binary operation $*$ is commutative and associative. Moreover, $e * f = f * e = f$ for every arithmetic function $f$. Thus the collection of all arithmetic functions forms a commutative monoid under $*$. It is natural to ask which arithmetic functions are invertible under $*$.
Theorem. Let $f$ be an arithmetic function. Then $f$ is invertible under Dirichlet convolution if and only if $f(1) \neq 0$ and in this case the inverse $f^{-1}$ is given recursively as $$f^{-1}(1) = \frac{1}{f(1)}, \qquad f^{-1}(n) = \sum_{\substack{d \, | \, n \\ d < n}} f^{-1}(d) f(n/d) \quad \text{for } n > 1.$$
The \emph{Euler’s totient function}, denoted $\varphi$ (or $\phi$), is defined as
\[
\varphi(n) = # {1 \leq k \leq n : (k, n) = 1}.
\]
In other words, $\varphi(n)$ is the number of reduced residue classes modulo $n$.
