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.

Contents

1 Completely additive
2 Examples
3 Multiplicative functions
4 References
5 See also

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) = 2^f(n).

References

Janko Bračič, Kolobar aritmetičnih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp 97 - 108) (MSC (2000) 11A25)

Additive function

Completely additive

Examples

Multiplicative functions

References

See also