multiplicatiiviset

Multiplicatiiviset, in Finnish mathematical terminology, describe properties or objects related to multiplication. In number theory, the term is often applied to arithmetic functions, which are functions f from the natural numbers to the complex (or real) numbers that preserve multiplicative structure in a certain sense.

One common use is for multiplicative functions. A function f: N → C is multiplicative if f(1) = 1 and f(mn) = f(m) f(n) whenever m and n are coprime (gcd(m,n) = 1). If the equality f(mn) = f(m) f(n) holds for all m and n, the function is completely multiplicative. Examples include the Möbius function μ(n), Euler’s totient function φ(n) (which is multiplicative but not completely multiplicative), and the divisor function d(n) (also written τ(n)). The constant function f(n) = 1 and the identity function f(n) = n are completely multiplicative; power functions f(n) = n^k are completely multiplicative for any fixed k.

A key consequence of multiplicativity is that the value of f on a general n can be determined from its values on prime powers. If n = p1^a1 · · · pr^ar, then f(n) = f(p1^a1) · · · f(pr^ar) for a multiplicative f, and for a completely multiplicative f one has f(p^k) = f(p)^k.

Multiplicativity also interacts with Dirichlet convolution: the set of multiplicative functions is closed under this operation, so if f and g are multiplicative, their Dirichlet convolution f * g is multiplicative as well.

Beyond arithmetic functions, the term multiplicative extends to concepts such as multiplicative monoids, multiplicative groups, and ring homomorphisms that preserve multiplication, all conveying a theme of structure-preserving multiplication.