Multiplicative function

A function defined on the natural numbers with the property that whenever a and b are coprime,

f(ab) = f(a)f(b).

Examples include Euler's φ function, the identity function, the functions d and σ defined by d(n) = the number of divisors of n and σ(n) = the sum of all the divisors of n, and many other functions of importance in number theory.

A multiplicative function is completely determined by its values on prime powers.

A function is said to be "completely multiplicative" if the property above holds even when a,b are not coprime. In this case the function is completely determined by its restriction to the prime numbers.