Properties
A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic. Thus, if n is a product of powers of distinct primes, say n = pa qb ..., then f(n) = f(pa) f(qb) ...
This property of multiplicative functions significantly reduces the need for computation, as in the following examples for n = 144 = 24 · 32:
- d(144) = 0(144) = 0(24)0(32) = (10 + 20 + 40 + 80 + 160)(10 + 30 + 90) = 5 · 3 = 15,
- (144) = 1(144) = 1(24)1(32) = (11 + 21 + 41 + 81 + 161)(11 + 31 + 91) = 31 · 13 = 403,
- *(144) = *(24)*(32) = (11 + 161)(11 + 91) = 17 · 10 = 170.
Similarly, we have:
- (144)=(24)(32) = 8 · 6 = 48
In general, if f(n) is a multiplicative function and a, b are any two positive integers, then
- f(a) · f(b) = f(gcd(a,b)) · f(lcm(a,b)).
Every completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.
Read more about this topic: Multiplicative Function
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