
Glossary: Prime Pages: Top 5000: 
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 nonzero.
If f(n) is multiplicative and we factor n into distinct primes as n=p_{1}^{a1}^{.} p_{2}^{a2}^{.} ...^{.}p_{k}^{ak}, then f(n) = f(p_{1}^{a1})^{.} f(p_{2}^{a2})^{.} ...^{.}f(p_{k}^{ak}).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
Chris K. Caldwell © 19992018 (all rights reserved)
