Multiplicativity

Multiplicativity is a property of functions defined on positive integers that reflects how the function interacts with multiplication. A function f is multiplicative if f(1) = 1 and f(ab) = f(a) f(b) whenever a and b are coprime. If the equation f(ab) = f(a) f(b) holds for all a and b without the coprimality condition, f is called completely multiplicative.

Common examples include the Euler totient function phi(n), which is multiplicative; the divisor function d(n) (the number of divisors), which is also multiplicative; and the Möbius function mu(n), which is multiplicative but not completely. The identity function id(n) = n and any power function n^k are completely multiplicative, as is the constant function 1. For a prime power p^k with k > 1, mu(p^k) = 0 while mu(p)^2 = 1, illustrating the distinction between multiplicative and completely multiplicative.

Multiplicativity is central in analytic number theory. For a multiplicative f, the Dirichlet series sum_{n>=1} f(n)/n^s often factors into an Euler product over primes: product over primes p of sum_{k>=0} f(p^k)/p^{ks}. This factorization underpins the appearance of the Riemann zeta function and various L-functions. The Dirichlet convolution of multiplicative functions is again multiplicative, enabling structured ways to build new arithmetic functions from existing ones.

Beyond number theory, multiplicativity also appears in algebra as the property of maps that preserve multiplication, such as ring homomorphisms, which satisfy f(xy) = f(x)f(y) for all elements x and y.