Multiplicative function

A multiplicative function is 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 the powers of prime numbers, a consequence of the fundamental theorem of arithmetic.

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