In number theory, an additive function is an arithmetic function f(n) of the positive integer n when a and b are coprime and:
- f(ab) = f(a) + f(b).
An additive function f(n) is said to be completely additive if f(ab) = f(a) + f(b) holds for all positive integers a and b, even when they are not coprime.
Every completely additive function is additive, but not vice versa.
Outside number theory, the term additive is usually used for all functions with the property f(ab) = f(a) + f(b) for all arguments a and b. This article discusses number theoretic additive functions.
Examples
Arithmetic functions which are completely additive are:
- A contraction of the logarithmic function on N.
- A function Ω(n), defined for every n ≥ 2 of total number of primes, which devide given positive integer n. We put also Ω(1) = 0. Some values:
- Ω(4) = 2
- Ω(27) = 3
- Ω(2,000) = 2
- Ω(2,001) = 3
- Ω(2,002) = 4
- Ω(2,003) = 1
- Ω(54,032,858,972,279) = 3
- Ω(54,032,858,972,302) = 6
- Ω(20,802,650,704,327,415) = 7
- ...
An example of an arithmetic function which is additive but not completely additive is:
- ω(n) = ∑p|n 1(n),
for every positive integer n, where sum runs over all different primes that do not devide n and 1(n) is a constant function, defined by 1(n) = 1. The ω function tells us how many different primes devide arbitrary positive integer n. Some values (compare with Ω(n)):
- ω(4) = 1
- ω(27) = 1
- ω(2,000) = 2
- ω(2,001) = 3
- ω(2,002) = 4
- ω(2,003) = 1
- ω(54,032,858,972,279) = 3
- ω(54,032,858,972,302) = 5
- ω(20,802,650,704,327,415) = 5
- ...
References
Sources:
- Janko Bračič, Kolobar aritmetičnih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp 97 - 108) (MSC (2000) 11A25)