Multiplikativ

Multiplikativ, in mathematics, describes a property of functions or operations that preserve multiplication. In the common context of arithmetic functions on the positive integers, a function f is called multiplicativ 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 positive integers m and n without the coprime restriction, the function is called completely multiplicative.

The distinction between multiplicative and completely multiplicative is important in number theory, as many functions satisfy one property but not the other. Classic examples include the Möbius function μ(n) and Euler’s totient function φ(n), both of which are multiplicative. Functions like f(n) = n^k (for any fixed k) are completely multiplicative, as f(mn) = (mn)^k = m^k n^k for all m, n. Dirichlet characters are another broad family of completely multiplicative functions.

Beyond arithmetic functions, the term also appears in algebra to describe maps that preserve multiplication. A function f between semigroups or rings is multiplicative if f(xy) = f(x) f(y) for all x, y in the domain. In ring theory, a ring homomorphism is a multiplicative map that also preserves addition.

In German-language mathematics, either "multiplikativ" or its English counterpart "multiplicative" is used to denote these ideas. The concept underpins many constructions, including Dirichlet convolution, multiplicative functions, and various product-preserving mappings across algebraic structures.