
Glossary: Prime Pages: Top 5000: 
A function f(n) defined on the positive integers
is completely multiplicative if
f(nm)=f(n)f(m)
for all pairs n and m (compare this with
multiplicative functions). Three simple examples
are f(n)=0, f(n)=1, and f(n)=n^{c}
(for a fixed positive value c).
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}.
Chris K. Caldwell © 19992018 (all rights reserved)
