multiplicativefunction

A multiplicative function is an arithmetic function f from the positive integers to the complex numbers that satisfies f(1) = 1 and f(mn) = f(m) f(n) whenever m and n are coprime. This property makes the function determined by its values at prime powers, since every n factors into primes as n = ∏ p^a, and f(n) = ∏ f(p^a).

A closely related notion is a completely multiplicative function, where the condition f(mn) = f(m) f(n) holds for all positive integers m and n, not just when they are coprime. For completely multiplicative functions, values on prime powers satisfy f(p^k) = f(p)^k.

Examples of multiplicative functions include the identity function id(n) = n, the divisor-counting function d(n) (or τ(n)) which counts the number of divisors, Euler’s totient φ(n), which counts integers up to n that are coprime to n, and the Möbius function μ(n). The Liouville function λ(n) is completely multiplicative, with λ(p^k) = (-1)^k.

Key properties include the fact that a multiplicative function is completely determined by its values on primes and their powers. The Dirichlet series associated with a multiplicative function factors as an Euler product: ∑_{n≥1} f(n)/n^s = ∏_p (1 + f(p)/p^s + f(p^2)/p^{2s} + …) for appropriate s. The class is closed under pointwise multiplication and under Dirichlet convolution, with the product or convolution of multiplicative functions again multiplicative.

See also: completely multiplicative, Euler product, Dirichlet convolution, arithmetic function.