Nonmultiplicative

In mathematics, the term *nonmultiplicative* refers to a property or function that does not satisfy the multiplicative property, meaning it does not preserve multiplication under composition or application. A multiplicative function is one where the product of two inputs yields the product of the function's values at those inputs, typically expressed as *f(ab) = f(a)f(b)* for all positive integers *a* and *b*. A nonmultiplicative function fails to adhere to this rule.

Many arithmetic functions are studied in number theory, and their multiplicative or nonmultiplicative nature is a key classification. For example, the identity function *f(n) = n* is multiplicative because *f(ab) = ab = f(a)f(b)*. In contrast, the sum-of-divisors function *σ(n)*, which counts the sum of all positive divisors of *n*, is nonmultiplicative. While it is *completely multiplicative* for square-free numbers, it does not generally satisfy *σ(ab) = σ(a)σ(b)* for all integers *a* and *b*. For instance, *σ(4) = 1 + 2 + 4 = 7*, but *σ(2)σ(2) = 3 × 3 = 9*, showing that *σ* is not multiplicative.

Nonmultiplicative functions often arise in number theory due to their dependence on specific properties of their arguments, such as divisibility or prime factorization. Some functions, like the Euler’s totient function *φ(n)*, are multiplicative but not completely multiplicative, meaning they satisfy *φ(ab) = φ(a)φ(b)* only when *a* and *b* are coprime. Functions that fail to satisfy even this weaker condition are strictly nonmultiplicative.

Understanding whether a function is multiplicative or nonmultiplicative helps in analyzing its behavior, particularly in divisibility and prime-related problems. Nonmultiplicative functions require more nuanced approaches, often relying on explicit formulas or recursive definitions rather than simple multiplicative rules.