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4.2: Additive Functions - Mathematics


Also important are the additive functions to which we will return in Chapter ??.

Definition 4.10

An additive function is a sequence such that (gcd(a, b) = 1) implies (f (ab) = f(a) + f(b)). A completely addititive function is one where the condition that (gcd(a, b) = 1) is not needed.

Definition 4.11

Let (omega (n)) denote the number of distinct prime divisors of (n) and let (Omega (n)) denote the number of prime powers that are divisors of (n). These functions are called the prime omega functions.

So if (n = prod_{i=1}^{s} p_{i}^{l_{i}}), then

[egin{array}{ccc} {omega_{n} = s}&{and}&{Omega (n) = sum_{i = 1}^{2} l_{i}} end{array} onumber]

The additivity of (omega) and the complete additivity of (Omega) should be clear.