Multiplicative function

Definition

Let $f$ be an arithmetic function: in other words, $f$ is a function from the set of natural numbers to a commutative unital ring $R$. We say that $f$ is multiplicative if it satisfies the following two conditions:

• $f(1)=1$.
• $f(mn)=f(m)f(n)$ for all pairs of relatively prime numbers $m,n\in \mathbb{N}$.

Facts

Determined by values at prime powers

A multiplicative function $f$ is determined completely by the values it takes at powers of primes. Further, the values taken by $f$ at prime powers are completely independent. In other words, any function from the set of prime powers to the commutative unital ring extends uniquely to a multiplicative function.

Dirichlet series

There is a nice Dirichlet series expression for multiplicative functions. Specifically, the Dirichlet series for a multiplicative function is a product of series for values at powers of each prime. If $f$ is a multiplicative function and $\mathbb{P}$ denotes the set of primes, we have:

${\sum }_{n\in \mathbb{N}}\frac{f(n)}{n^s}={\prod }_{p\in \mathbb{P}}(\sum _{k=0}^{\mathrm{\infty }}\frac{f(p^k)}{p^{ks}})$.

Relation with other properties

Stronger properties

• Completely multiplicative function

Incomparable properties

• Divisibility-preserving function

Effect of operations

Dirichlet product

Further information: Multiplicative functions form a group under Dirichlet product

Under the Dirichlet product, the multiplicative functions form a group. In other words, the Dirichlet product of two multiplicative functions is multiplicative, and the inverse of a multiplicative function with respect to the Dirichlet product is also a multiplicative function.

Pointwise product

The pointwise product of two multiplicative functions is a multiplicative function. Note that a multiplicative function is invertible with respect to the pointwise product if and only if its value at every natural number is invertible, in which case its inverse is also a multiplicative function. In particular, a multiplicative function that is zero at any natural number cannot be invertible with respect to the pointwise product.

Composition

If $f,g$ are multiplicative functions such that $g$ takes only positive integer values, it is not necessary that $f\circ g$ be a multiplicative function. The reason is that $g(m)$ and $g(n)$ need not be relatively prime even if $m$ and $n$ are relatively prime.

However, it is true that if $f$ is completely multiplicative and $g$ is multiplicative with positive integer values, then $f\circ g$ is multiplicative. Further information: Composite of completely multiplicative function and multiplicative function is multiplicative