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multiplicative function (another Prime Pages' Glossary entries) |
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Glossary:
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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
Chris K. Caldwell © 1999-2012 (all rights reserved)
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