Additive function
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Different definitions exist depending on the specific field of application. Traditionally, an additive function is a function that preserves the addition operation:
- f(x+y) = f(x)+f(y)
for any two elements x and y in the domain.
In number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever a and b are coprime, the function of the product is the sum of the functions:
- f(ab) = f(a) + f(b).
Outside number theory, the term additive may also be used for all functions with the property f(ab) = f(a) + f(b) for all arguments a and b.
The remainder of this article discusses number theoretic additive functions, using the second definition. For a specific case of the first definition see additive polynomial. Note also that any homomorphism f between Abelian groups is "additive" by the first definition.
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Completely additive
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.
Examples
Arithmetic functions which are completely additive are:
- The restriction of the logarithmic function to N, a0(n) - the sum of primes dividing n, sometimes called sopfr(n). We have a0(20) = a0(22 · 5) = 2 + 2+ 5 = 9. Some values: (SIDN A001414).
- a0(4) = 4
- a0(27) = 9
- a0(144) = a0(24 · 32) = a0(24) + a0(32) = 8 + 6 = 14
- a0(2,000) = a0(24 · 53) = a0(24) + a0(53) = 8 + 15 = 23
- a0(2,003) = 2003
- a0(54,032,858,972,279) = 1240658
- a0(54,032,858,972,302) = 1780417
- a0(20,802,650,704,327,415) = 1240681
- ...
- a1(n) - the sum of the distinct primes dividing n, sometimes called sopf(n). We have a1(1) = 0, a1(20) = 2 + 5 = 7. Some more values: (SIDN A008472)
- a1(4) = 2
- a1(27) = 3
- a1(144) = a1(24 · 32) = a1(24) + a1(32) = 2 + 3 = 5
- a1(2,000) = a1(24 · 53) = a1(24) + a1(53) = 2 + 5 = 7
- a1(2,001) = 55
- a1(2,002) = 33
- a1(2,003) = 2003
- a1(54,032,858,972,279) = 1238665
- a1(54,032,858,972,302) = 1780410
- a1(20,802,650,704,327,415) = 1238677
- ...
- The function Ω(n), defined as the total number of prime factors of n, counting multiple factors multiple times. It is often called "Big Omega function".This implies Ω(1) = 0 since 1 has no prime factors. Some more values: (SIDN A001222)
- Ω(4) = 2
- Ω(27) = 3
- Ω(144) = Ω(24 · 32) = Ω(24) + Ω(32) = 4 + 2 = 6
- Ω(2,000) = Ω(24 · 53) = Ω(24) + Ω(53) = 4 + 3 = 7
- 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), defined as the total number of different prime factors of n. Some values (compare with Ω(n)) (SIDN A001221)
- ω(4) = 1
- ω(27) = 1
- ω(144) = ω(24 · 32) = ω(24) + ω(32) = 1 + 1 = 2
- ω(2,000) = ω(24 · 53) = ω(24) + ω(53) = 1 + 1 = 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
- ...
Multiplicative functions
From any additive function f(n) it is easy to create a related multiplicative function g(n) i.e. with the property that whenever a and b are coprime we have:
- g(ab) = g(a) × g(b).
One such example is g(n) = 2f(n).
References
- Janko Bračič, Kolobar aritmetičnih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp 97 - 108) (MSC (2000) 11A25)