Multiplicative function

An arithmetic function satisfying f(ab)=f(a)f(b) whenever the positive integers a and b are coprime. Its values on prime powers determine its values on all positive integers.

Arithmetic function: A function from the positive integers to a number system, studied through its values on integers. Multiplicative functions are a structured subclass of arithmetic functions.
Completely multiplicative function: An arithmetic function satisfying f(ab)=f(a)f(b) for all positive integers a and b. It imposes the same product rule even when the inputs share prime factors.
Möbius function: The arithmetic function that is zero on integers divisible by a squared prime and otherwise equals (−1)^k for k distinct prime factors. Its prime-factor definition gives a canonical example of multiplicativity.
Dirichlet character: A periodic, completely multiplicative function used to encode residue classes modulo an integer. These functions turn congruence conditions into multiplicative coefficients for Dirichlet series.
Additive function: An arithmetic function satisfying f(ab)=f(a)+f(b) whenever a and b are coprime. It replaces the product of function values with a sum.
Coprime integers: Two integers are coprime when their greatest common divisor is 1. The multiplicative identity is required precisely for coprime inputs.
Dirichlet convolution: The operation (f*g)(n)=Σ_{d|n} f(d)g(n/d) on arithmetic functions. Convolving two multiplicative functions produces another multiplicative function.
Euler's totient function: The function φ(n) counting the integers from 1 to n that are coprime to n. Coprime residue classes combine to give φ(ab)=φ(a)φ(b) when a and b are coprime.
Dirichlet L-function: The Dirichlet series L(s,χ)=Σχ(n)n^−s associated with a Dirichlet character χ. Complete multiplicativity of χ gives this series an Euler product.
Quasimultiplicative function: An arithmetic function satisfying f(ab)=f(a)f(b)/f(1) for coprime positive integers a and b, with f(1) nonzero. This variant allows a non-unit value at 1 while retaining a normalized product rule.

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