multiplicative function

A function f(n) defined on the positive integers is multiplicative if f(nm)=f(n)f(m) whenever n and m are relatively prime. Clearly f(1) must be 0 or 1. If f(1)=0, then f(n)=0 for all positive integers n. So some authors require that f(1) be non-zero.

If f(n) is multiplicative and we factor n into distinct primes as n=p1a1. p2a2. ....pkak, then

f(n) = f(p1a1). f(p2a2). ....f(pkak).

Finally, if f(n) is multiplicative, then so is the function F(n) = sum of f(i) (where the sum is taken over the divisors i of n).

See Also: CompletelyMultiplicative, EulersPhi

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