Multiplicative function

Definition

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

• .
• for all pairs of relatively prime numbers .

Facts

Determined by values at prime powers

A multiplicative function is determined completely by the values it takes at powers of primes. Further, the values taken by 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 is a multiplicative function and denotes the set of primes, we have: .

Effect of operations

Dirichlet product

Further information: Multiplicative functions form abelian group under Dirichlet product

Under the Dirichlet product, the multiplicative functions form an abelian 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 are multiplicative functions such that takes only positive integer values, it is not necessary that be a multiplicative function. The reason is that and need not be relatively prime even if and are relatively prime.

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